Abstract. Coalescing compact binaries are thought to be involved in a wide variety of astrophysical phenomena. In particular, they are important sources of gravitational radiation for both ground-based and space-based laser-interferometer detectors, and they may be sources of supernova explosions or gamma-ray bursts. Mergers of two white dwarfs may produce neutron stars with peculiar properties, including perhaps millisecond radio pulsars sometimes accompanied by planets (as observed in PSR 1257+12). According to a widely held belief, the coalescence of two neutron stars should produce a rapidly rotating black hole surrounded by an accretion disk or torus, but this is by no means certain. This review paper focuses on the final hydrodynamic coalescence and merger of double neutron stars and double white dwarfs, and addresses the question of the nature of the final merger products.