This is a partial answer specifically about government kanji lists (prior answers did not go into detail).
MEXT (文部省) and MOJ (法務省) publish official kanji lists (漢字表) like 常用漢字表 (MEXT) and 人名用漢字票 (MOJ). The former is a list of about 2000 kanji "for every day use" (常用) that schools teach and children are expected to learn in school. The second is a supplementary list of about 1000 kanji "for use in names" (人名用), that are not judged to be particularly common outside use in names. The union of these sets represents kanji that are "legally recognized" (can be used in legal documents without issue, etc).
Note that native speakers generally learn to recognize/read many kanji beyond what is in the government lists (the lists account for about 3000 "official" kanji). (See note below on numbers beyond "government recognized" kanji)
MEXT further divides the 常用 list. A subset of 1006 kanji is called the 教育漢字表, or "educational kanji list", which is further separated by grade level and is taught in elementary school. First graders learn 80 kanji (plus hiragana and katakana). Second graders learn another 160, third and fourth graders learn 200 a piece, fifth graders learn 185, and sixth graders learn 181.
These lists are not the first versions. The government has published kanji lists in the past, but the older ones have been replaced by the newest iterations.
Aside: How many kanji are there, and how many do native speakers know? It is hard to give an estimate. To give an idea of how many kanji exist that are relevant to Japanese, Wikipedia suggests that a dictionary containing 50,000 unique kanji is considered comprehensive (but of course many of these are outdated or specialist-only), and that JIS standards call for computer encodings of 13,000 kanji, with the main range including 6,000. I queried the JMDIC data set, which contains spellings of Japanese words representing close to 6,000 kanji, with close to 3,000 of these being in words marked "common". The Kanji Kentei Pre-1 and 1 levels have low pass rates even among adult native speakers, but they require very in-depth reading and hand-writing knowledge of ~3,000 and ~6,000 respectively. To me it seems reasonable to say that a typical native speaker will have some familiarity with readings or meanings of between 3,000 to 13,000 kanji, but a more precise number would be very sensitive to a definition of "familiarity" and background of the native speaker.