This monograph is a self-contained introduction to the geometry of Riemann Surfaces of constant curvature 1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces.The first part of the book is written in textbook form at the graduate level, with few requisites other than background in either differential geometry or complex Riemann surface theory. It begins with an account of the FenchelNielsen approach to Teichmüller Space. Hyperbolic trigonometry and Bers partition theorem (with a new proof which yields explicit bounds) are shown to be simple but powerful tools in this context. The second part of the book is a self-contained introduction to the spectrum of the Laplacian based on head equations. The approach chosen yields a simple proof that compact Riemann surfaces have the same eigenvalues if and only if they have the same length spectrum. Later chapters deal with recent developments on isospectrality, Sunadas construction, a simplified proof of Wolperts theorem, and an estimate of the number of pairwise isospectral non-isometric examples which depends only on genus.Research workers and graduate students interested in compact Riemann surfaces will find here a number of useful tools and insights to apply to their investigations.

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