The paper presents a review of the basic concepts of the Logical Analysis of Data (LAD), along with a series of discrete optimization models associated to the implementation of various components of its general methodology, as well as an outline of applications of LAD to medical problems. The combinatorial optimization models described in the paper represent variations on the general theme of set covering, including some with nonlinear objective functions. The medical applications described include the development of diagnostic and prognostic systems in cancer research and pulmonology, risk assessment among cardiac patients, and the design of biomaterials.