A book with k pages consists of a straight line (the spine) and k half-planes (the pages), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). The k-page crossing number νk(G) of a graph G is the minimum number of crossings in a k-page drawing of G. In this paper we investigate the k-page crossing numbers of complete graphs. We use semidefinite programming techniques to give improved lower bounds on νk(Kn) for various values of k. We also use a maximum satisfiability reformulation to obtain a computer-aided calculation of the exact value of νk(Kn) for several values of k and n. Finally, we investigate the best construction known for drawing Kn in k pages, calculate the resulting number of crossings, and discuss this upper bound in light of the new results reported in this paper.