Conventionally, a Josephson junction is an ubiquitous quantum device formed by a weak link between a pair of superconductors. In this work, we demonstrate the dc Josephson effect in mesoscopic junctions of more than two superconducting terminals. We report fabrication and characterization of the 3- and 4-terminal Josephson junctions built in a top-down fashion from hybrid semiconductor-superconductor InAs/Al epitaxial heterostructures. In general, the critical current of an N-terminal junction is an (N-1)-dimensional hypersurface in the space of bias currents, which can be reduced to a set of critical current contours (CCCs). The CCC is a key ground state characteristic of a multi-terminal Josephson junction, which is readily available from regular electron transport measurements. We investigate nontrivial modifications of the CCC's geometry in response to electrical gating, magnetic field, and phase bias. All observed effects are described by the scattering formulation of the Josephson effect generalized to the case of N>2. Our observations indicate superconducting phase coherence between all the terminals which establishes the Josephson effect in mesoscopic junctions of more than two superconductors. Such multi-terminal junctions could find their applications in a broad range of fields from topologically protected quantum computation to quantum metrology and others.