  We present some results on the proportion of permutations of length $n$
containing certain mesh patterns as $n$ grows large, and give exact enumeration
results in some cases. In particular, we focus on mesh patterns where entire
rows and columns are shaded. We prove some general results which apply to mesh
patterns of any length, and then consider mesh patterns of length four. An
important consequence of these results is to show that the proportion of
permutations containing a mesh pattern can take a wide range of values between
$0$ and $1$.
