  We present a new, all-sky quasar catalog, Quaia, that samples the largest
comoving volume of any existing spectroscopic quasar sample. The catalog draws
on the 6,649,162 quasar candidates identified by the Gaia mission that have
redshift estimates from the space observatory's low-resolution BP/RP spectra.
This initial sample is highly homogeneous and complete, but has low purity, and
18% of even the bright ($G<20.0$) confirmed quasars have discrepant redshift
estimates ($|\Delta z/(1+z)|>0.2$) compared to those from the Sloan Digital Sky
Survey (SDSS). In this work, we combine the Gaia candidates with unWISE
infrared data (based on the Wide-field Infrared Survey Explorer survey) to
construct a catalog useful for cosmological and astrophysical quasar studies.
We apply cuts based on proper motions and Gaia and unWISE colors, reducing the
number of contaminants by $\sim$4$\times$. We improve the redshifts by training
a $k$-nearest neighbors model on SDSS redshifts, and achieve estimates on the
$G<20.0$ sample with only 6% (10%) catastrophic errors with $|\Delta
z/(1+z)|>0.2$ ($0.1$), a reduction of $\sim$3$\times$ ($\sim$2$\times$)
compared to the Gaia redshifts. The final catalog has 1,295,502 quasars with
$G<20.5$, and 755,850 candidates in an even cleaner $G<20.0$ sample, with
accompanying rigorous selection function models. We compare Quaia to existing
quasar catalogs, showing that its large effective volume makes it a highly
competitive sample for cosmological large-scale structure analyses. The catalog
is publicly available at https://zenodo.org/records/10403370.
