We present the results of the Extremely Luminous Quasar Survey in the 3π survey of the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS; PS1). This effort applies the successful quasar selection strategy of the Extremely Luminous Survey in the Sloan Digital Sky Survey footprint (∼12,000 deg2) to a much larger area (∼21,486 deg2). This spectroscopic survey targets the most luminous quasars (M 1450 ≤ -26.5 m i ≤ 18.5) at intermediate redshifts (z ≥ 2.8). Candidates are selected based on a near-infrared JKW2 color cut using WISE AllWISE and 2MASS photometry to mainly reject stellar contaminants. Photometric redshifts (z reg) and star-quasar classifications for each candidate are calculated from near-infrared and optical photometry using the supervised machine learning technique random forests. We select 806 quasar candidates at z reg ≥ 2.8 from a parent sample of 74,318 sources. After exclusion of known sources and rejection of candidates with unreliable photometry, we have taken optical identification spectra for 290 of our 334 good PS-ELQS candidates. We report the discovery of 190 new z ≥ 2.8 quasars and an additional 28 quasars at lower redshifts. A total of 44 good PS-ELQS candidates remain unobserved. Including all known quasars at z ≥ 2.8, our quasar selection method has a selection efficiency of at least 77%. At lower declinations, -30 ≤ decl. ≤ 0, we approximately treble the known population of extremely luminous quasars. We provide the PS-ELQS quasar catalog with a total of 592 luminous quasars (m i ≤ 18.5, z ≥ 2.8). This unique sample will not only be able to provide constraints on the volume density and quasar clustering of extremely luminous quasars, but also offers valuable targets for studies of the intergalactic medium.