This article reviews the recent progress on the subject of semi-Dirac fermions, two dimensional quasiparticles that disperse quadratically, as Galilean invariant particles, or linearly, as massless relativistic particles, depending on their direction of motion. These particles exist at a phase boundary set by the continuous change in the connectivity of Fermi surfaces, known as topological Lifshitz transitions. The basic properties and the current experimental evidence of the existence of these particles in both synthetic lattices and quantum materials are presented. The many-body problem of semi-Dirac fermions is discussed from a theoretical perspective with an eye to physical observables of relevance to experiments.