In 1979, Bernhart and Kainen conjectured that graphs of fixed genus g ≥ 1 have unbounded pagenumber. In this paper, it is proven that genus g graphs can be embedded in O ( g ) pages, thus disproving the conjecture. An Ω( g 1/2 ) lower bound is also derived. The first algorithm in the literature for embedding an arbitrary graph in a book with a non-trivial upper bound on the number of pages is presented. First, the algorithm computes the genus g of a graph using the algorithm of Filotti, Miller, Reif (1979), which is polynomial-time for fixed genus. Second, it applies an optimal-time algorithm for obtaining an O ( g )-page book embedding. Separate book embedding algorithms are given for the cases of graphs embedded in orientable and nonorientable surfaces. An important aspect of the construction is a new decomposition theorem, of independent interest, for a graph embedded on a surface. Book embedding has application in several areas, two of which are directly related to the results obtained: fault-tolerant VLSI and complexity theory.