We present an overview of the single- and multiple-sector results of the Science Processing Operations Center (SPOC) transit search in the primary Transiting Exoplanet Survey Satellite (TESS) mission. TESS was designed to survey bright stars in the greater Solar neighborhood in search of transiting exoplanets. Data were acquired at a 2-minute cadence for 16,000-20,000 pre-selected target stars in each 28-day observation sector and processed in the SPOC pipeline at NASA Ames Research Center. The photometry pipeline produced a systematic error corrected light curve for each target star. Light curves were searched for transiting planet signatures by sector for all target stars, and separately for target stars observed in multiple sectors. Potential transit signals for which the transiting planet detection threshold was exceeded and a series of transit consistency tests were passed are referred to as Threshold Crossing Events (TCEs). We highlight the full TCE population and the population of SPOC TCEs that were later identified as TESS Objects of Interest (TOIs). Characteristics of the TCE populations implied by limb-darkened transiting planet model fits are also presented. SPOC pipeline data products are delivered to the Mikulski Archive for Space Telescopes (MAST; archive.stsci.edu/tess) for access by the community. Funding for the TESS Mission has been provided by the NASA Science Mission Directorate.