Neutron stars and black holes, the densest objects in the Universe, are formed in the graveyards of stars. In a neutron star matter takes exotic forms, which is a major puzzle for modern nuclear physics. In black
holes, matter is converted into an extreme form of spacetime curvature, such that a black hole is completely described by its mass, angular momentum and electric charge. Binaries of neutron stars or black holes can be formed in a number of astronomical scenarios. Due to their compact concentrations of mass-energy and relativistic motions, such systems are strong sources of gravitational waves, which will carry away the orbital energy, causing the two stars to draw closer and eventually to merge with each other. The merger of such compact objects are among the most energetic events in the Universe, where a few percentage of the mass of the objects is converted into gravitational energy according to Einstein’s famous formula E = mc^2. Such objects are also some of the best ‘cosmic laboratories’: gravitational wave observations from these objects will lead to precision tests of theories of gravity as well as provide a wealth of information relevant to fundamental physics, astrophysics and cosmology. This talk is aimed to provide a non-technical introduction to the physics and astrophysics of compact binaries.