It is proved that if a Schur superalgebra is not semisimple, then it is neither cellular nor quasi-hereditary (Theorem 2), and it has infinite global dimension (Corollary 18). The algebra $S(m|n,r)$ with $m,n \ge 1$ is semisimple if and only if $p$, the characteristic of the ground field, is zero or greater than $r$, or when $m=n=1$ and $p$ does not divide $r$.