  A $0$-$1$ matrix $M$ is saturating for a $0$-$1$ matrix $P$ if $M$ does not
contain a submatrix that can be turned into $P$ by changing some $1$ entries to
$0$ entries, and changing an arbitrary $0$ to $1$ in $M$ introduces such a
submatrix in $M$. In saturation problems for $0$-$1$ matrices we are interested
in estimating the minimum number of $1$ entries in an $m \times n$ matrix that
is saturating for $P$, in terms of $m$ and $n$.
  In other words, we wish to give good estimates for the saturation function of
$P$. Recently, Brualdi and Cao initiated the study of saturation problems in
the context of $0$-$1$ matrices.
  We extend their work in several directions. We prove that every $0$-$1$
forbidden matrix has its saturation function either in $\Theta(1)$ or
$\Theta(n)$ in the case when we restrict ourselves to square saturating
matrices. Then we give a partial answer to a question posed by Brualdi and Cao
about the saturation function of $J_k$, which is obtained from the identity
matrix $I_k$ by putting the first row after the last row. Furthermore, we
exhibit a $5\times 5$ permutation matrix with the saturation function bounded
from the above by a fixed constant. We complement this result by identifying
large classes of $0$-$1$ matrices with linear saturation function. Finally, we
completely resolve the related semisaturation problem as far as the constant
vs. linear dichotomy is concerned.
