The quantum cosmology of two-dimensional dilaton-gravity models is investigated. A class of models is mapped onto the constrained oscillator-ghost-oscillator model. A number of exact and approximate solutions to the corresponding Wheeler-DeWitt equation are presented. A wider class of minisuperspace models that can be solved in this fashion is identified. Supersymmetric extensions to the induced gravity theory and the bosonic string theory are then considered and closed-form solutions to the associated quantum constraints are derived. The possibility of applying the third-quantization procedure to two-dimensional dilaton-gravity is briefly discussed.

28 pages, latex