We generalize the concept of a differential-equation eigenvalue problem from linear to nonlinear differential equations. For nonlinear differential equations the eigenfunctions are separatrix curves and the eigenvalues are the initial conditions that give rise to separatrix curves. The Painlevé transcendents serve as examples of nonlinear eigenvalue problems, and nonlinear semiclassical techniques are devised to calculate the large-n behavior of the nth eigenvalue. This behavior is found by reducing the Painlevé equation to the linear Schrödinger equation associated with a non-Hermitian PT-symmetric Hamiltonian. The notion of a nonlinear differential-equation eigenvalue problem extends beyond the Painlevé transcendents to huge classes of nonlinear differential equations. Some of these equations exhibit new and interesting eigenvalue behavior such as spectra having fine structure.