  We use the Simba cosmological galaxy formation simulation to investigate the
relationship between major mergers ($\leq$ 4:1), starbursts, and galaxy
quenching. Mergers are identified via sudden jumps in stellar mass $M_*$ well
above that expected from in situ star formation, while quenching is defined as
going from specific star formation rate sSFR$>t_{H}^{-1}$ to
sSFR$<0.2t_{H}^{-1}$, where $t_{H}$ is the Hubble time. At $z\approx 0-3$,
mergers show $\sim\times 2-3$ higher SFR than a mass-matched sample of
star-forming galaxies, but globally represent $\leq 1\%$ of the cosmic SF
budget. At low masses, the increase in SFR in mergers is mostly attributed to
an increase in the $H_2$ content, but for $M_*\geq 10^{10.5} M_{\odot}$ mergers
also show an elevated star formation efficiency suggesting denser gas within
merging galaxies. The merger rate for star-forming galaxies shows a rapid
increase with redshift $\propto (1+z)^{3.5}$, but the quenching rate evolves
much more slowly, $\propto (1+z)^{0.9}$; there are insufficient mergers to
explain the quenching rate at $z\leq 1.5$. Simba first quenches galaxies at
$z\geq 3$, with a number density in good agreement with observations. The
quenching timescales $\tau_q$ are strongly bimodal, with `slow' quenchings
($\tau_q \sim 0.1t_{H}$) dominating overall, but `fast' quenchings ($\tau_q\sim
0.01 t_H$) dominating in $M_*\sim 10^{10}-10^{10.5}M_{\odot}$ galaxies, likely
induced by Simba's jet-mode black hole feedback. The delay time distribution
between mergers and quenching events suggests no physical connection to either
fast or slow quenching. Hence, Simba predicts that major mergers induce
starbursts, but are unrelated to quenching in either fast or slow mode.
