  We survey some major contributions to Riemann's moduli space and
Teichm{\"u}ller space. Our report has a historical character, but the stress is
on the chain of mathematical ideas. We start with the introduction of Riemann
surfaces, and we end with the discovery of some of the basic structures of
Riemann's moduli space and Teichm{\"u}ller space. We point out several facts
which seem to be unknown to many algebraic geometers and analysts working in
the theory. The period we are interested in starts with Riemann, in 1851, and
ends in the early 1960s, when Ahlfors and Bers confirmed that Teichm{\"u}ller's
results were correct.This paper was written for the book "Lipman Bers, a life
in Mathematics," edited by Linda Keen , Irwin Kra and Rubi Rodriguez (Amercian
Mathematical Society, 2015). It is dedicated to the memory of Lipman Bers who
was above all a complex analyst and spent a large part of his life and energy
working on the analytic structure of Teichm{\"u}ller space. His work on
analysis is nevertheless inseparable from geometry and topology. In this
survey, we highlight the relations and the logical dependence between this work
and the works of Riemann, Poincar{\'e}, Klein, Brouwer, Siegel,
Teichm{\"u}ller, Weil, Grothendieck and others. We explain the motivation
behind the ideas. In doing so, we point out several facts which seem to be
unknown to many Teichm{\"u}ller theorists.
