  We present the results of the Extremely Luminous Quasar Survey in the $3\pi$
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
($\sim12,000\,\rm{deg}^2$) to a much larger area
($\sim\rm{21486}\,\rm{deg}^2$). This spectroscopic survey targets the most
luminous quasars ($M_{1450}\le-26.5$; $m_{i}\le18.5$) at intermediate redshifts
($z\ge2.8$). Candidates are selected based on a near-infrared JKW2 color cut
using WISE AllWISE and 2MASS photometry to mainly reject stellar contaminants.
Photometric redshifts ($z_{\rm{reg}}$) 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 $z_{\rm{reg}}\ge2.8$ from a parent sample of 74318 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 $z\ge2.8$ quasars
and an additional 28 quasars at lower redshifts. A total of 44 good PS-ELQS
candidates remain unobserved. Including all known quasars at $z\ge2.8$, our
quasar selection method has a selection efficiency of at least $77\%$. At lower
declinations $-30\le\rm{Decl.}\le0$ we approximately triple the known
population of extremely luminous quasars. We provide the PS-ELQS quasar catalog
with a total of 592 luminous quasars ($m_{i}\le18.5$, $z\ge2.8$). 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.
