Studies of the most luminous quasars at high redshift directly probe the evolution of the most massive black holes in the early universe and their connection to massive galaxy formation. However, extremely luminous quasars at high redshift are very rare objects. Only wide-area surveys have a chance to constrain their population. The Sloan Digital Sky Survey (SDSS) has so far provided the most widely adopted measurements of the quasar luminosity function at . However, a careful re-examination of the SDSS quasar sample revealed that the SDSS quasar selection is in fact missing a significant fraction of quasars at the brightest end. We identified the purely optical-color selection of SDSS, where quasars at these redshifts are strongly contaminated by late-type dwarfs, and the spectroscopic incompleteness of the SDSS footprint as the main reasons. Therefore, we designed the Extremely Luminous Quasar Survey (ELQS), based on a novel near-infrared JKW2 color cut using Wide-field Infrared Survey Explorer mission (WISE) AllWISE and 2MASS all-sky photometry, to yield high completeness for very bright ( ) quasars in the redshift range of . It effectively uses random forest machine-learning algorithms on SDSS and WISE photometry for quasar-star classification and photometric redshift estimation. The ELQS will spectroscopically follow-up ∼230 new quasar candidates in an area of ∼12,000 deg2 in the SDSS footprint to obtain a well-defined and complete quasar sample for an accurate measurement of the bright-end quasar luminosity function (QLF) at . In this paper, we present the quasar selection algorithm and the quasar candidate catalog.