In 1976, the antifungal hydroxamic antibiotic trichostatin A (TSA) was isolated from the metabolites of the bacterium Streptomyces hygroscopicus. It took 14 years to find that TSA has an effect on the proliferation and differentiation of mammalian cells by inhibiting histone deacetylase (HDAC) activity. By 2015, a single database containing the structures of about 1050 synthetic and 400 natural HDAC inhibitors had been created. Currently, five inhibitors are approved for use as anticancer agents, with dozens more compounds undergoing clinical trials. However, the implementation of new agents is severely hampered by their multidirectional action and the severity of side effects. New strategies are needed to overcome these problems, including the development of inhibitors targeting a specific class of HDACs. In addition to the most important characteristics of histone deacetylases and their natural inhibitors, this review considers current approaches to the design of selective HDAC inhibitors and the methods used to test them.