We consider the eigenvalue problem for the Sturm-Liouville equation with the Dirichlet boundary condition at the left end of the interval and with the boundary condition containing arbitrary analytical functions of the eigenparameter at the right end. The inverse problem, which consists in recovering the Sturm-Liouville potential from a part of the spectrum, will be analysed. We wil discuss necessary and sufficient conditions of uniqueness and a constructive solution of this problem, its global solvability, local solvability, and stability. In addition, applications of this problem to studying partial inverse problems on intervals and on graphs, and also to the inverse transmission eigenvalue problem will be considered.