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The ALMaQUEST (ALMA-MaNGA QUEnching and STar formation) survey is a program with spatially-resolved $^{12}$CO(1-0) measurements obtained with the Atacama Large Millimeter Array (ALMA) for 46 galaxies selected from the Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) DR15 optical integral-field spectroscopic survey. The aim of the ALMaQUEST survey is to investigate the dependence of star formation activity on the cold molecular gas content at kpc scales in nearby galaxies. The sample consists of galaxies spanning a wide range in specific star formation rate (sSFR), including starburst (SB), main-sequence (MS), and green valley (GV) galaxies. In this paper, we present the sample selection and characteristics of the ALMA observations, and showcase some of the key results enabled by the combination of spatially-matched stellar populations and gas measurements. Considering the global (aperture-matched) stellar mass, molecular gas mass, and star formation rate of the sample, we find that the sSFR depends on both the star formation efficiency (SFE) and the molecular gas fraction ($f_{\rm H_{2}}$), although the correlation with the latter is slightly weaker. Furthermore, the dependence of sSFR on the molecular gas content (SFE or $f_{\rm H_{2}}$) is stronger than that on either the atomic gas fraction or the molecular-to-atomic gas fraction, albeit with the small HI sample size. On kpc scales, the variations in both SFE and $f_{\rm H_{2}}$ within individual galaxies can be as large as 1-2 dex thereby demonstrating that the availability of spatially-resolved observations is essential to understand the details of both star formation and quenching processes.