  We measure the CO-to-H$_2$ conversion factor ($\alpha_\mathrm{CO}$) in 37
galaxies at 2~kpc resolution, using dust surface density inferred from
far-infrared emission as a tracer of the gas surface density and assuming a
constant dust-to-metals ratio. In total, we have $\sim790$ and $\sim610$
independent measurements of $\alpha_\mathrm{CO}$ for CO (2-1) and (1-0),
respectively. The mean values for $\alpha_\mathrm{CO~(2-1)}$ and
$\alpha_\mathrm{CO~(1-0)}$ are $9.3^{+4.6}_{-5.4}$ and
$4.2^{+1.9}_{-2.0}~M_\odot~pc^{-2}~(K~km~s^{-1})^{-1}$, respectively. The
CO-intensity-weighted mean for $\alpha_\mathrm{CO~(2-1)}$ is 5.69, and 3.33 for
$\alpha_\mathrm{CO~(1-0)}$. We examine how $\alpha_\mathrm{CO}$ scales with
several physical quantities, e.g.\ star-formation rate (SFR), stellar mass, and
dust-mass-weighted average interstellar radiation field strength
($\overline{U}$). Among them, $\overline{U}$, $\Sigma_{\rm SFR}$, and
integrated CO intensity ($W_\mathrm{CO}$) have the strongest anti-correlation
with spatially resolved $\alpha_\mathrm{CO}$. We provide linear regression
results to \aco for all quantities tested. At galaxy integrated scales, we
observe significant correlations between $\alpha_\mathrm{CO}$ and
$W_\mathrm{CO}$, metallicity, $\overline{U}$, and $\Sigma_{\rm SFR}$. We also
find that the normalized $\alpha_\mathrm{CO}$ decreases with stellar mass
surface density ($\Sigma_\star$) in the high surface density regions
($\Sigma_\star\geq100~{\rm M_\odot~pc^{-2}}$), following the power-law
relations $\alpha_\mathrm{CO~(2-1)}\propto\Sigma_\star^{-0.5}$ and
$\alpha_\mathrm{CO~(1-0)}\propto\Sigma_\star^{-0.2}$. The power-law index is
insensitive to the assumed dust-to-metals ratio. (abridged)
