  We construct and study the theory of relative quasimaps in genus zero, in the
spirit of Gathmann. When $X$ is a smooth toric variety and $Y$ is a smooth very
ample hypersurface in $X$, we produce a virtual class on the moduli space of
relative quasimaps to $(X,Y)$, which we use to define relative quasimap
invariants. We obtain a recursion formula which expresses each relative
invariant in terms of invariants of lower tangency, and apply this formula to
derive a quantum Lefschetz theorem for quasimaps, expressing the restricted
quasimap invariants of $Y$ in terms of those of $X$. Finally, we show that the
relative $I$-function of Fan-Tseng-You coincides with a natural generating
function for relative quasimap invariants, providing mirror-symmetric
motivation for the theory.
