  Combinatorics on multisets is used to deduce new upper and lower bounds on
the number of numerical semigroups of each given genus, significantly improving
existing ones. In particular, it is proved that the number $n_g$ of numerical
semigroups of genus $g$ satisfies $2F_{g}\leq n_g\leq 1+3\cdot 2^{g-3}$, where
$F_g$ denotes the $g$th Fibonacci number.
