The research program in seismic exploration in progress in the Mathematics Dept. of the Univ. of Denver is described. The mathematical formulations employed by this group are outlined and results of computer implementation are depicted. Ongoing research is also presented. The main method we have developed is identified here by the term 'velocity inversion.' This is a method based upon perturbation theory, leading to a linear integral equation for the variation in velocity from a fixed reference value. Applied to high-frequency data, the method yields a map of the interfaces in the interior of the earth, as well as estimates of reflection strength at those interfaces. A new non-perturbation approach to the velocity inversion problem is also presented. This is a high-frequency method which accounts for two major nonlinear effects, namely, transmission losses and refraction. The method also has some similarity to wave-equation migration in that a wave equation is derived for the ensemble of backscattered signals, scaled by time, and with initial data having their support on the reflecting surfaces and in known proportion to the reflection strength.