These lecture notes represent the content of a
course taught at the institute of Mathematical Sciences,
New York University, during the academic year 1957-1958.
The notes reflect the lectures as actually given. A
proper course on Riemann surfaces would need about twice
the time that I had at my disposal. For that reason,
many important topics were not covered; in particular,
Abel's theorem, the Jacobi inversion problem, classifi￾cation of open Riemann surfaces, qussiconformal mappings,
the problem of moduli.
I owe a debt of gratitude to Professor Robert
Osserman, who replaced me on several occasions; Lectures
14, 23, 24 and 25 were given by him. I am also very
grateful to Mr. Richard Pollack and Miss Esther Rodlitz,
who prepared these notes.