Rambam

General

1. The Maharashdam believes that since the Rambam saw the other opinions on each issue and still decided the way we did, we should follow his rulings (analogous to the Radbaz and the Rif). On the other hand, a number of Ashkenazi authorities feel that the Rambam is a minority in the face of the Baalei HaTosafot.
2. The Rambam was a descendant of David HaMelech and Rabbi Yehudah HaNassi.
3. One cannot pose questions from Tosafot's logic against the Rambam. Gavra aGavra KaRamit?!
4. The Rivash and Rosh warn that one who attempts to rule on practical matters solely from Mishneh Torah without understanding its Talmudic context will likely mistake Assur for Muttar and Muttar for Assur, thinking he understands the material. At the same time, the Rambam's explicit intention was to write the Halacha so clearly that one need not be bogged down by the confusing Sugya and need only Mishneh Torah to know how to act, so some Acharonim, such as the Ohr HaChaim and Baal HaTanya are less concerned. While that is true, the Rambam himself explains in a letter that he did so for people for whom understanding the Gemara was beyond their reach.
5. The Rambam is drawn extremely after the Rif and seldom disagrees with him, so one should not assume so unless there are already a great number of Rishonim for him to side with.

Mishneh Torah

General Approach

1. In Mishneh Torah, the Rambam focuses on relaying rulings explicated in the Gemara in a lucid way, while leaving out rulings that can only be inferred but aren't stated explicitly.
2. The Rambam's style is to write only what it says in the Gemara and not the interpretation, but his intention is to for the interpretation of the Gemara to be applied to his words, as well, though some disagree.
3. It's abnormal for the Rambam to omit a din dealt with explicitly in the Talmud and instead write a different idea not mentioned in the Talmud that implies the one mentioned explicitly.
4. Counts of the number of Perakim, Halachot, Mitzvot, etc are provided in Mishneh Torah to facilitate memorization, not to imply additional ideas.
5. The Rambam will rule like a Halacha found in the Sifrei if it's not contradicted by the Talmud.
6. Every word of the Rambam is written with tremendous precision and exactness, enough for one to infer from his words as one would from the Gemara itself. Similarly, the Maharalbach warns that one shouldn't rush to reject the Rambam for coming off as against the Gemara, but should rather pin the in-congruence on our own lack of understanding.
7. One cannot infer from the presentation of a ruling in one section of Mishneh Torah and its absence in another seemingly appropriate section that it only applies in one and not the other, as the Rambam's goal is to present the Talmud's rulings in their wording in their appropriate place. For example, a Halacha in Terumot that might also fit in Maachalot Assurot cannot be assumed to only apply in Terumot and not Maachalot Assurot given its presence in the former and absence in the latter. This is, however, a valid tool in reading other Poskim, such as the Tur.
8. It's debatable if Halachot whose source Gemarot have unresolved textual variants (Safek Girsaot) are omitted by the Rambam.

Writing Style

1. The word Assur is used even when referring to Dinim MiDeoraita that would warrant Malkot.
2. "MiPi HaShmua" refers to a Halacha whose source is not explicit in the Torah, and "MiPi HaKabbalah" refers to Halachot learned from Pesukim in Neviim.
3. The term "MiDivrei Sofrim" can refer to Dinim MiDeRabbanan, as well as any Din MiDeoraita not stated explicitly in the Torah, such as one derived via the thirteen Middot, because without the Chachamim it would not be understood.
4. Consistently marking each Din as a Gezeirah or Halacha LeMoshe MiSinai is not a priority for the Rambam: sometime he mentions the Din's classification, while other times he does not.
5. The Rivash argues that the Rambam will refer to a Takkanat Chachamim as "Torah," such as in "Lo Hikpida Torah" in reference to the Ne'emanut of an Ed Echad, while others disagree.
6. When the Gemara presents more specific arguments predicated on accepting one of two possible assumptions - "Im Timtzi Lomar" - the Geonim and Rambam understand that the Halacha assumes like that assumption. Some say this is only true if the Gemara did not leave the very same uncertainty pending with a "Tiku" elsewhere, while others argue the opposite. The two levels of the Im Timtzi Lomar cannot be interchangeable, meaning if assuming A over B and then asking C over D is equivalent to assuming C over D and then asking A over B, this rule does not apply for the Rambam. The words must also be explicit in the text of the Gemara, so two subsequent questions, one building on the previous, would not be subject to this rule. The Poskim dispute if this is only true when the Amora in the Gemara himself uses these words to navigate the situation or if it's even true when the omniscient narrator of the Sugya does so externally of the Amora's words. Some say that the Rambam will not follow Im Timtzi Lomar if the Gemara explicitly assumes the opposite idea to be true.
7. If the Gemara presents two opposite understandings and then rejects one, the Rambam will formulate it in a way that accepts the accepted one and implies the rejection of the rejected one.
8. Often times, answers presented in the Gemara in rejection of a difficulty are omitted by the Rambam in favor of the simple meaning of the Mishnah, Baraita, or Amoraic statement.
9. It is not uncommon for the Rambam to pin a ruling on a Pasuk not mentioned in or even rejected by the Gemara, because he felt it to be simpler or more acceptable, especially if there's no practical difference.
10. In a few places, the Rambam employs a Talmudic phrase to mean something different from what it does in the Talmud.

Connections to Other Works

1. Though the Rambam never relies in Mishneh Torah on what he already wrote in Peirush HaMishnayot, he does rely on what he wrote in earlier sections of Mishneh Torah, and perhaps even later ones, or, at least, in that chapter.
2. When a contradiction is found between the Perush HaMishnayot and the Mishneh Torah, the Halacha follows Mishneh Torah, which was written later and as a set of rulings, not a set of elucidations of the Mishnah.

Misc.

1. The entire Mishneh Torah took ten years of non-stop effort to produce.
2. The Roshei Tevot of the first four words of Mishneh Torah, "יסוד היסודות ועמוד החכמה" spells the Shem Havaya.

Sefer HaMitzvot

1. The focus of Sefer HaMitzvot is not to determine which Mitzvot are part of the 613, not to present a comprehensive of accurate representation of their Halachot and details. Therefore, when studying Sefer HaMitzvot, questioning the count of Mitzvot is wholly acceptable, but questioning the details of the Mitzvot is not. The Mishneh Torah was written to address those details, and, even if there are distinct differences, it's always possible that he changed his mind over time.
2. There's a debate regarding if specific Rishonim such as the Samag, Maggid Mishneh, and Migdal Oz saw the Sefer HaMitzvot, as it wasn't translated from Arabic to Hebrew for some time. Later Acharonim culled lists of instances in which each of them cites the Sefer HaMitzvot to dispel those claims.

Perush HaMishnah

1. The goal of the Rambam's commentary is clearly to decide Halacha, as he decides Halacha in each Mishnah.
2. In Perush HaMishnayot, wherever there is an opinion that misleadingly seems to be the minority one but is, in fact, the one the Halacha follows, the Rambam will buttress that opinion by writing of its truth or singularity or the like to indicate that the Halacha does indeed follow it.
3. The Perush HaMishnah (also known as "Maor" or "Hisraj") was originally written in Arabic and then translated into Hebrew by various people.
4. The Ramban did not have the Introduction to Perush HaMishnayot available to him, as Hebrew translations of the Arabic work were not yet available in Spain, according to the Yad Malachi, who posits that had they been available, the Ramban would not have been as quick to argue. The Rashba and Terumat HaDeshen did not have the Perush HaMishnah available at all either.
5. Sometimes our translation of Perush HaMishnayot will refer to a din as being MiDeRabbanan, but it's probably just a mistaken translation of "MiDivrei Sofrim."

Acceptance

1. By well into the fifteenth century, many Sephardic communities around the world accepted the Rambam as their Halachic leader.

Other Works

1. Shu"t Pe'er HaDor is a compilation of his Teshuvot translated into Hebrew from the Arabic original.
2. He also compiled a Sefer on Klalei HaTalmud, but it is no longer extant.
3. Moreh HaNevuchim was translated from Arabic to Hebrew and the source of much controversy regarding the study of philosophy. This Chida suggests he wrote it so that people who were more interested in philosophy should give weight to what he has to say in Mishneh Torah, as well.
4. The Rambam wrote a commentary on much of Talmud Bavli, as indicated by his introduction to Perush HaMishnayot, but only fragments of it have survived, including a commentary on Rosh HaShanah and the beginning of Shabbat.
5. None of his commentary on the Talmud Yerushalmi is extant.

Misc.

1. The Ramban spoke of the Rambam with great esteem.
2. Some thought that the Ri Migash was the Rambam's teacher, but he was more likely Rabbeinu Maimon's teacher, whom the Rambam encountered as a child. Regardless, the Rambam does refer to the Rif and Ri Migash as "Rabbotai."
3. Some claim that the Rambam was forcibly converted to Islam, but such a thought is highly questionable.