What is known and not known about rational points of modular curves? What are some good references?

Dear Dick, I am going to interpret rational points to mean points over $\mathbb Q$. Given this, Mazur's Eisenstein ideal paper, his rational isogenies paper, and various surveys he wrote around that time (midtolate 1970s) give a good description (and also prove the most of the key results). Ogg also wrote surveys in the early 70s, making conjectures which Mazur then proved, which are very helpful. You'll easily find them on mathscinet (as you will with Mazur's papers and surveys). Best wishes, Matt 

