In this paper, we consider recent progress on the multiplicity of the solution set of various nonlinear boundary value problems. We first consider the semilinear Dirichlet problem and discuss some recent progress on a conjecture of the authors arising from an earlier paper of Ambrosetti and Prodi. We then consider other global bifurcation results for the Dirichlet problem, and finally some results for parabolic and hyperbolic semilinear equations.