This book deals with several bound state and scattering problems in quantum theory. There exist numerous unsolved bound state problems in both non- relativistic and relativistic theories. Solving them either analytically or numerically can be of immense importance both from theoretical as well as from experimental points of view. This is the task undertaken in this book. Several interesting bound state problems are solved exactly. Some problems are solved using well-established methods. Wherever possible, the results of our calculations are compared with those obtained by using different methods. Comments are made about the physical situations that correspond to the potentials chosen for our study. Apart from bound states, the phenomenon of scattering is an important tool to understand the structure of particles and forces between them. Many of the properties like charge, spin, size etc. of microscopic particles are better understood with the help of scattering experiments. In this book we discus relativistic corrections to the scattering cross section using the well-known phase shift analysis. An interesting paradoxial situation that arises for strong potentials is also discussed.