Existence and geometric structure of metrics on surfaces which extremize eigenvalues

This is an exposition of the research area around our lecture at the 60^th anniversary conference of IMPA which was held in October of 2012. It is a survey of results which have been obtained over many years concerning sharp upper bounds on the first eigenvalue of a surface, either with or without boundary, in terms or area or boundary length and the surface topology. It is mostly expository, but contains a new coarse upper bound for non-orientable surfaces with boundary. It also contains a classical reformulation of recent results in 10].