Movies

ID  Genre
28  Action
12  Adventure
16  Animation
35  Comedy
80  Crime
99  Documentary
18  Drama
10751   Family
14  Fantasy
36  History
27  Horror
10402   Music
9648    Mystery
10749   Romance
878 Science Fiction
10770   TV Movie
53  Thriller
10752   War
37  Western

TV Show

ID  Genre
10759   Action & Adventure
16  Animation
35  Comedy
80  Crime
99  Documentary
18  Drama
10751   Family
10762   Kids
9648    Mystery
10763   News
10764   Reality
10765   Sci-Fi & Fantasy
10766   Soap
10767   Talk
10768   War & Politics
37  Western

@prince_m said:
How to get the list genres available
Since the docs are down, does anybody know how to get the list of genres available

See answer above.

and how to also get a list of movies or tv shows based a genre?

In the explanation below there are links to the Docs that, as you said, are currently down.
Discover - Filter by Genre
First you need to have the list of Genres with their IDs.
The list of Genre IDs, for Movies and TV, can be found here:
get/genre/movie/list - https://developers.themoviedb.org/3/genres/get-movie-list
get/genre/tv/list - https://developers.themoviedb.org/3/genres/get-tv-list

For example, the Comedy Genre ID for movies is 35.

Knowing the Genre ID you want, make an API Request of type:
get / discover / movie
https://developers.themoviedb.org/3/discover/movie-discover
get / discover / tv
https://developers.themoviedb.org/3/discover/tv-discover
And use the with_genres and/or without_genres parameter.
More than one genre can be searched using the comma or pipe feature

Also note that a number of filters support being comma (,) or pipe (|) separated.
Comma's are treated like an AND and query while Pipe's are an OR.

AND (with Comma) &with_genres=35,37,80
OR (with Pipe) &with_genres=35|37|80
You cannot mix AND with OR. Either one or the other.

You can use any other filter simultaneously.
&year=, &sort_by=, &with_cast= etc...

Example

https://api.themoviedb.org/3/discover/movie?api_key=THE_KEY&language=pt-BR&page=1&with_genres=35,37,80  
https://api.themoviedb.org/3/discover/tv?api_key=THE_KEY&language=pt-BR&page=1&with_genres=35,37,80