We provide the exact solution of several variants of simple models of the zipping transition of two bound polymers, such as occurs in DNA/RNA, in two and three dimensions using pairs of directed lattice paths. In three dimensions the solutions are written in terms of complete elliptic integrals. We analyse the phase transition associated with each model giving the scaling of the partition function. We also extend the models to include a pulling force between one end of the pair of paths, which competes with the attractive monomer-monomer interactions between the polymers.