We review some perturbative and nonperturbative aspects of topological string theory on the Calabi-Yau manifolds X_p=O(-p) +O(p-2) ->P^1. These are exactly solvable models of topological string theory which exhibit a nontrivial yet simple phase structure, and have a phase transition in the universality class of pure two--dimensional gravity. They don't have conventional mirror description, but a mirror B model can be formulated in terms of recursion relations on a spectral curve typical of matrix model theory. This makes it possible to calculate nonperturbative, spacetime instanton effects in a reliable way, and in particular to characterize the large order behavior of string perturbation theory.