Phosphorescent organic light-emitting diodes (OLEDs) are leading candidates for next-generation displays and solid-state lighting technologies. Much of the academic and commercial pursuits in phosphorescent OLEDs have been dominated by Ir(III) complexes. Over the past decade recent developments have enabled square planar Pt(II) and Pd(II) complexes to meet or exceed the performance of Ir complexes in many aspects. In particular, the development of N-heterocyclic carbene-based emitters and tetradentate cyclometalated Pt and Pd complexes have significantly improved the emission efficiency and reduced their radiative lifetimes making them competitive with the best reported Ir complexes. Furthermore, their unique and diverse molecular design possibilities have enabled exciting photophysical attributes including narrower emission spectra, excimer -based white emission, and thermally activated delayed fluorescence. These developments have enabled the fabrication of efficient and "pure" blue OLEDs, single-doped white devices with EQEs of over 25% and high CRI, and device operational lifetimes which show early promise that square planar metal complexes can be stable enough for commercialization. These accomplishments have brought Pt complexes to the forefront of academic research. The molecular design strategies, photophysical characteristics, and device performance resulting from the major advancements in emissive Pt and Pd square planar complexes are discussed.