The processes used by academic and industrial scientists to discover new drugs have recently experienced a true renaissance, with many new and exciting techniques being developed over the past 5–10 years alone. Drug design and discovery, and the search for new safe and well-tolerated compounds, as well as the ineffectiveness of existing therapies, and society’s insufficient knowledge concerning the prophylactics and pharmacotherapy of the most common diseases today, comprise a serious challenge. This can influence not only the quality of human life, but also the health of whole societies, which became evident during the COVID-19 pandemic. In general, the process of drug development consists of three main stages: drug discovery, preclinical development using cell-based and animal models/tests, clinical trials on humans and, finally, forward moving toward the step of obtaining regulatory approval, in order to market the potential drug. In this review, we will attempt to outline the first three most important consecutive phases in drug design and development, based on the experience of three cooperating and complementary academic centers of the Visegrád group; i.e., Medical University of Lublin, Poland, Masaryk University of Brno, Czech Republic, and Comenius University Bratislava, Slovak Republic.