  We derive the first constraints on the time delay distribution of binary
black hole (BBH) mergers using the LIGO-Virgo Gravitational-Wave Transient
Catalog GWTC-2. Assuming that the progenitor formation rate follows the star
formation rate (SFR), the data favor that $43$--$100\%$ of mergers have delay
times $<4.5$ Gyr (90% credibility). Adopting a model for the metallicity
evolution, we derive joint constraints for the metallicity-dependence of the
BBH formation efficiency and the distribution of time delays between formation
and merger. Short time delays are favored regardless of the assumed metallicity
dependence, although the preference for short delays weakens as we consider
stricter low-metallicity thresholds for BBH formation. For a $p(\tau) \propto
\tau^{-1}$ time delay distribution and a progenitor formation rate that follows
the SFR without metallicity dependence, we find that $\tau_\mathrm{min}<2.2$
Gyr, whereas considering only the low-metallicity $Z < 0.3\,Z_\odot$ SFR,
$\tau_\mathrm{min} < 3.0$ Gyr (90% credibility). Alternatively, if we assume
long time delays, the progenitor formation rate must peak at higher redshifts
than the SFR. For example, for a $p(\tau) \propto \tau^{-1}$ time delay
distribution with $\tau_\mathrm{min} = 4$ Gyr, the inferred progenitor rate
peaks at $z > 3.9$ (90% credibility). Finally, we explore whether the inferred
formation rate and time delay distribution vary with BBH mass.
