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הלכות תשובה א,ד; ב,ט-יא; ד,ג
In the beginning of Hilkhot Teshuvah, Maimonides states that teshuvah atones for all sins. After establishing this general principle, he goes on to develop the details. Below are some "juicy" halakhot that complicate this principle in interesting and (I think) compelling ways.
In hevruta, pick a text, read and reflect! I suggest you use this format that I learned from Dr. Laura Yares:
What is something you affirm in the text?
What is something you challenge?
What is an implication that you see?
In 1:4, he states that the timeline and steps necessary for atonement differ based on the gravity of the transgression.

(ד) אַף עַל פִּי שֶׁהַתְּשׁוּבָה מְכַפֶּרֶת עַל הַכּל וְעַצְמוֹ שֶׁל יוֹם הַכִּפּוּרִים מְכַפֵּר. יֵשׁ עֲבֵרוֹת שֶׁהֵן מִתְכַּפְּרִים לִשְׁעָתָן וְיֵשׁ עֲבֵרוֹת שֶׁאֵין מִתְכַּפְּרִים אֶלָּא לְאַחַר זְמַן. כֵּיצַד. עָבַר אָדָם עַל מִצְוַת עֲשֵׂה שֶׁאֵין בָּהּ כָּרֵת וְעָשָׂה תְּשׁוּבָה אֵינוֹ זָז מִשָּׁם עַד שֶׁמּוֹחֲלִין לוֹ, וּבְאֵלּוּ נֶאֱמַר (ירמיה ג כב) "שׁוּבוּ בָּנִים שׁוֹבָבִים אֶרְפָּה מְשׁוּבֹתֵיכֶם" וְגוֹ'. עָבַר עַל מִצְוַת לֹא תַּעֲשֶׂה שֶׁאֵין בָּהּ כָּרֵת וְלֹא מִיתַת בֵּית דִּין וְעָשָׂה תְּשׁוּבָה, תְּשׁוּבָה תּוֹלָה וְיוֹם הַכִּפּוּרִים מְכַפֵּר, וּבְאֵלּוּ נֶאֱמַר (ויקרא טז ל) "כִּי בַיּוֹם הַזֶּה יְכַפֵּר עֲלֵיכֶם". עָבַר עַל כְּרֵתוֹת וּמִיתוֹת בֵּית דִּין וְעָשָׂה תְּשׁוּבָה, תְּשׁוּבָה וְיוֹם הַכִּפּוּרִים תּוֹלִין וְיִסּוּרִין הַבָּאִין עָלָיו גּוֹמְרִין לוֹ הַכַּפָּרָה. וּלְעוֹלָם אֵין מִתְכַּפֵּר לוֹ כַּפָּרָה גְּמוּרָה עַד שֶׁיָּבוֹאוּ עָלָיו יִסּוּרִין, וּבְאֵלּוּ נֶאֱמַר (תהילים פט לג) "וּפָקַדְתִּי בְשֵׁבֶט פִּשְׁעָם וּבִנְגָעִים עֲוֹנָם". בַּמֶּה דְּבָרִים אֲמוּרִים בְּשֶׁלֹּא חִלֵּל אֶת הַשֵּׁם בְּשָׁעָה שֶׁעָבַר אֲבָל הַמְחַלֵּל אֶת הַשֵּׁם אַף עַל פִּי שֶׁעָשָׂה תְּשׁוּבָה וְהִגִּיעַ יוֹם הַכִּפּוּרִים וְהוּא עוֹמֵד בִּתְשׁוּבָתוֹ וּבָאוּ עָלָיו יִסּוּרִין אֵינוֹ מִתְכַּפֵּר לוֹ כַּפָּרָה גְּמוּרָה עַד שֶׁיָּמוּת. אֶלָּא תְּשׁוּבָה יוֹם הַכִּפּוּרִים וְיִסּוּרִין שְׁלָשְׁתָּן תּוֹלִין וּמִיתָה מְכַפֶּרֶת שֶׁנֶּאֱמַר (ישעיה כב יד) "וְנִגְלָה בְאָזְנָי ה' צְבָאוֹת" וְגוֹ' (ישעיה כב יד) "אִם יְכֻפַּר הֶעָוֹן הַזֶּה לָכֶם עַד תְּמֻתוּן":

(4) Even though Teshuvah atones for all [sins] and the essence of Yom Kippur brings atonement, [there are different levels of sin and hence, differences in the degree of atonement.] There are sins that can be atoned for immediately and other sins which can only be atoned for over the course of time. What is implied?
If a person violates a positive command which is not punishable by karet and repents, he will not leave that place before he is forgiven. Concerning these sins, [Jeremiah 3:22] states: "Return, faithless children! I will heal your rebellious acts."
If a person violates a prohibition that is not punishable by karet or execution by the court and repents, Teshuvah has a tentative effect and Yom Kippur brings atonement as [Leviticus, loc. cit. states "This day will atone for you."
If a person violates [sins punishable by] karet or execution by the court and repents, Teshuvah and Yom Kippur have a tentative effect and the sufferings which come upon him complete the atonement. He will never achieve complete atonement until he endures suffering for concerning these [sins, Psalms 89:33] states: "I will punish their transgression with a rod."***
When does the above apply: When the desecration of God's name is not involved in the transgression. However, a person who desecrated God's name, even though he repented, Yom Kippur arrived while he continued his repentance, and he experienced suffering, will not be granted complete atonement until he dies. The three: repentance, Yom Kippur, and suffering have a tentative effect and death atones as [Isaiah 22:14] states: "It was revealed in my ears [by] the Lord of Hosts, surely this iniquity will not be atoned for until you die."

*** For the Rambam, this suffering was likely physical suffering.
In 2:9, Rambam asserts that for transgressions בין אדם לחברו, one does not achieve atonement until they have asked for, and received forgiveness from the person they harmed. These halakhot describe this process and its implications. 2:10 discusses the implications of this for the person who was harmed; 2:11 describes the process in a case where the person who was harmed has died.
2:9 describes the process for asking for forgiveness;

(ט) אֵין הַתְּשׁוּבָה וְלֹא יוֹם הַכִּפּוּרִים מְכַפְּרִין אֶלָּא עַל עֲבֵרוֹת שֶׁבֵּין אָדָם לַמָּקוֹם כְּגוֹן מִי שֶׁאָכַל דָּבָר אָסוּר אוֹ בָּעַל בְּעִילָה אֲסוּרָה וְכַיּוֹצֵא בָּהֶן. אֲבָל עֲבֵרוֹת שֶׁבֵּין אָדָם לַחֲבֵרוֹ כְּגוֹן הַחוֹבֵל אֶת חֲבֵרוֹ אוֹ הַמְקַלֵּל חֲבֵרוֹ אוֹ גּוֹזְלוֹ וְכַיּוֹצֵא בָּהֶן אֵינוֹ נִמְחַל לוֹ לְעוֹלָם עַד שֶׁיִּתֵּן לַחֲבֵרוֹ מַה שֶּׁהוּא חַיָּב לוֹ וִירַצֵּהוּ. אַף עַל פִּי שֶׁהֶחֱזִיר לוֹ מָמוֹן שֶׁהוּא חַיָּב לוֹ צָרִיךְ לְרַצּוֹתוֹ וְלִשְׁאל מִמֶּנּוּ שֶׁיִּמְחל לוֹ. אֲפִלּוּ לֹא הִקְנִיט אֶת חֲבֵרוֹ אֶלָּא בִּדְבָרִים צָרִיךְ לְפַיְּסוֹ וְלִפְגֹּעַ בּוֹ עַד שֶׁיִּמְחל לוֹ. לֹא רָצָה חֲבֵרוֹ לִמְחל לוֹ מֵבִיא לוֹ שׁוּרָה שֶׁל שְׁלֹשָׁה בְּנֵי אָדָם מֵרֵעָיו וּפוֹגְעִין בּוֹ וּמְבַקְּשִׁין מִמֶּנּוּ. לֹא נִתְרַצָּה לָהֶן מֵבִיא לוֹ שְׁנִיָּה וּשְׁלִישִׁית. לֹא רָצָה מְנִיחוֹ וְהוֹלֵךְ לוֹ וְזֶה שֶׁלֹּא מָחַל הוּא הַחוֹטֵא. וְאִם הָיָה רַבּוֹ הוֹלֵךְ וּבָא אֲפִלּוּ אֶלֶף פְּעָמִים עַד שֶׁיִּמְחל לוֹ:

(9) Teshuvah and Yom Kippur only atone for sins between man and God; for example, a person who ate a forbidden food or engaged in forbidden sexual relations, and the like. However, sins between man and man; for example, someone who injures a colleague, curses a colleague, steals from him, or the like will never be forgiven until he gives his colleague what he owes him and appeases him.


[It must be emphasized that] even if a person restores the money that he owes [the person he wronged], he must appease him and ask him to forgive him.
Even if a person only upset a colleague by saying [certain] things, he must appease him and approach him [repeatedly] until he forgives him.
If his colleague does not desire to forgive him, he should bring a group of three of his friends and approach him with them and request [forgiveness]. If [the wronged party] is not appeased, he should repeat the process a second and third time. If he [still] does not want [to forgive him], he may let him alone and need not pursue [the matter further]. On the contrary, the person who refuses to grant forgiveness is the one considered as the sinner.
[The above does not apply] if [the wronged party] was one's teacher. [In that instance,] a person should continue seeking his forgiveness, even a thousand times, until he forgives him.

2:10 describes the obligations of the harmed person:

(י) אָסוּר לָאָדָם לִהְיוֹת אַכְזָרִי וְלֹא יִתְפַּיֵּס אֶלָּא יְהֵא נוֹחַ לִרְצוֹת וְקָשֶׁה לִכְעֹס וּבְשָׁעָה שֶׁמְּבַקֵּשׁ מִמֶּנּוּ הַחוֹטֵא לִמְחל מוֹחֵל בְּלֵב שָׁלֵם וּבְנֶפֶשׁ חֲפֵצָה. וַאֲפִלּוּ הֵצֵר לוֹ וְחָטָא לוֹ הַרְבֵּה לֹא יִקֹּם וְלֹא יִטֹּר. וְזֶהוּ דַּרְכָּם שֶׁל זֶרַע יִשְׂרָאֵל וְלִבָּם הַנָּכוֹן. אֲבָל הָעוֹבְדֵי כּוֹכָבִים עַרְלֵי לֵב אֵינָן כֵּן אֶלָּא (וְעֶבְרָתָן) [וְעֶבְרָתוֹ] שְׁמָרָה נֶצַח. וְכֵן הוּא אוֹמֵר עַל הַגִּבְעוֹנִים לְפִי שֶׁלֹּא מָחֲלוּ וְלֹא נִתְפַּיְּסוּ וְהַגִּבְעֹנִים לֹא מִבְּנֵי יִשְׂרָאֵל הֵמָּה:

(10) It is forbidden for a person to be cruel and refuse to be appeased. Rather, he should be easily pacified, but hard to anger. When the person who wronged him asks for forgiveness, he should forgive him with a complete heart and a willing spirit. Even if he aggravated and wronged him severely, he should not seek revenge or bear a grudge.
This is the path of the seed of Israel and their upright spirit. In contrast, the insensitive gentiles do not act in this manner. Rather, their wrath is preserved forever. Similarly, because the Gibeonites did not forgive and refused to be appeased, [II Samuel 21:2] describes them, as follows: "The Gibeonites are not among the children of Israel."

2:11 describes what happens if the person who was harmed has died.

(יא) הַחוֹטֵא לַחֲבֵרוֹ וּמֵת חֲבֵרוֹ קֹדֶם שֶׁיְּבַקֵּשׁ מְחִילָה מֵבִיא עֲשָׂרָה בְּנֵי אָדָם וּמַעֲמִידָן עַל קִבְרוֹ וְיֹאמַר בִּפְנֵיהֶם חָטָאתִי לַה' אֱלֹהֵי יִשְׂרָאֵל וְלִפְלוֹנִי זֶה שֶׁכָּךְ וְכָךְ עָשִׂיתִי לוֹ. וְאִם הָיָה חַיָּב לוֹ מָמוֹן יַחֲזִירוֹ לַיּוֹרְשִׁים. לֹא הָיָה יוֹדֵעַ לוֹ יוֹרְשִׁין יַנִּיחֶנּוּ בְּבֵית דִּין וְיִתְוַדֶּה:

(11) If a person wronged a colleague and the latter died before he could ask him for forgiveness, he should take ten people and say the following while they are standing before the colleague's grave: "I sinned against God, the Lord of Israel, and against this person by doing the following to him...."
If he owed him money, he should return it to his heirs. If he is unaware of the identity of his heirs, he should place [the sum] in [the hands of] the court and confess.

In chapter 4, Rambam describes different categories of transgression that are exceptions, in one way or another, to the general rules of teshuvah.
4:3 lists categories of transgression for which full atonement is impossible because the person who transgressed can't (fully) identify the person who was harmed and therefore, can't ask for forgiveness.

(ג) וּמֵהֶם חֲמִשָּׁה דְּבָרִים הָעוֹשֶׂה אוֹתָן אִי אֶפְשָׁר לוֹ שֶׁיָּשׁוּב בִּתְשׁוּבָה גְּמוּרָה לְפִי שֶׁהֵם עֲוֹנוֹת שֶׁבֵּין אָדָם לַחֲבֵרוֹ וְאֵינוֹ יוֹדֵעַ חֲבֵרוֹ שֶׁחָטָא לוֹ כְּדֵי שֶׁיַּחֲזִיר לוֹ אוֹ יִשְׁאַל מִמֶּנּוּ לִמְחל לוֹ. וְאֵלּוּ הֵן. א) הַמְקַלֵּל אֶת הָרַבִּים וְלֹא קִלֵּל אָדָם יָדוּעַ כְּדֵי שֶׁיִּשְׁאַל מִמֶּנּוּ כַּפָּרָה. ב) וְהַחוֹלֵק עִם גַּנָּב, שֶׁלְּפִי שֶׁאֵינוֹ יוֹדֵעַ גְּנֵבָה זוֹ שֶׁל מִי הִיא אֶלָּא הַגַּנָּב גּוֹנֵב לָרַבִּים וּמֵבִיא לוֹ וְהוּא לוֹקֵחַ. וְעוֹד שֶׁהוּא מַחֲזִיק יַד הַגַּנָּב וּמַחְטִיאוֹ. ג) וְהַמּוֹצֵא אֲבֵדָה וְאֵינוֹ מַכְרִיז עָלֶיהָ עַד שֶׁיַּחֲזִירָהּ לִבְעָלֶיהָ, לְאַחַר זְמַן כְּשֶׁיַּעֲשֶׂה תְּשׁוּבָה אֵינוֹ יוֹדֵעַ לְמִי יַחֲזִיר. ד) וְהָאוֹכֵל שׁוֹר עֲנִיִּים וִיתוֹמִים וְאַלְמָנוֹת. אֵלּוּ בְּנֵי אָדָם אֻמְלָלִין הֵם וְאֵינָן יְדוּעִים וּמְפֻרְסָמִים וְגוֹלִים מֵעִיר לְעִיר וְאֵין לָהֶם מַכִּיר כְּדֵי שֶׁיֵּדַע שׁוֹר זֶה שֶׁל מִי הוּא וְיַחֲזִירֶנּוּ לוֹ. ה) וְהַמְקַבֵּל שֹׁחַד לְהַטּוֹת דִּין אֵינוֹ יוֹדֵעַ עַד הֵיכָן הִגִּיעַ הַטָּיָה זוֹ וְכַמָּה הִיא כֹּחָהּ כְּדֵי שֶׁיַּחֲזִיר, שֶׁהַדָּבָר יֵשׁ לוֹ רַגְלַיִם. וְעוֹד שֶׁהוּא מַחֲזִיק יַד זֶה וּמַחְטִיאוֹ:

(3) Among these [24] are five [transgressions] for which it is impossible for the person who commits them to repent completely. They are sins between man and man, concerning which it is impossible to know the person whom one sinned against in order to return [what is owed him] or ask for his forgiveness. They are:
a) One who curses the many without cursing a specific individual from whom he can request forgiveness;
b) One who takes a share of a thief's [gain], for he does not know to whom the stolen article belongs. The thief steals from many, brings him [his share], and he takes it. Furthermore, he reinforces the thief and causes him to sin;
c) One who finds a lost object and does not announce it [immediately] in order to return it to its owners. Afterwards, when he desires to repent, he will not know to whom to return the article;
d) One who eats an ox belonging to the poor, orphans, or widows. These are unfortunate people, who are not well-known or recognized by the public. They wander from city to city and thus, there is no one who can identify them and know to whom the ox belonged in order that it may be returned to him.
e) One who takes a bribe to pervert judgment. He does not know the extent of the perversion or the power [of its implications] in order to pay the [people whom he wronged], for his judgment had a basis. Furthermore, [by taking a bribe], he reinforces the person [who gave it] and causes him to sin.