This book describes the Hamilton-Jacobi formalism of quantum mechanics, which allowscomputation of eigenvalues of quantum mechanical potential problems without solving for thewave function. The examples presented include exotic potentials such as quasi-exactly solvablemodels and Lame an dassociated Lame potentials. A careful application of boundary conditionsoffers an insight into the nature of solutions of several potential models. Advancedundergraduates having knowledge of complex variables and quantum mechanics will find thisas an interesting method to obtain the eigenvalues and eigen-functions. The discussion oncomplex zeros of the wave function gives intriguing new results which are relevant foradvanced students and young researchers. Moreover, a few open problems in research arediscussed as well, which pose a challenge to the mathematically oriented readers.