We present a new class of quantum field theories which are exactly solvable. The theories are generated by introducing Pauli-Villars (PV) fermionic and bosonic fields with masses degenerate with the physical positive metric fields. An algorithm is given to compute the spectrum and corresponding eigensolutions. We also give the operator solution for a particular case and use it to illustrate some of the tenets of light-cone quantization. Since the solutions of the solvable theory contain ghost quanta, these theories are unphysical. However, the existence of an exact solution provides an important check on the implementation of PV-regulated discretized light-cone quantization (DLCQ). In the limit of exact mass degeneracy of the ghost and physical fields, the numerical DLCQ solutions are constrained to reduce to the explicit forms we give here. We also discuss how perturbation theory in the difference between the masses of the physical and PV particles could be developed, thus generating physical theories. The existence of explicit solutions of the solvable theory also allows one to study the relationship between the equal-time and light-cone vacua and eigensolutions.