  Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely
generated $R$-module. We say $M$ has maximal depth if there is an associated
prime $\mathfrak{p}$ of $M$ such that depth $M=\dim R/\mathfrak{p}$. In this
paper, we study finitely generated modules with maximal depth. It is shown that
the maximal depth property is preserved under some important module operations.
Generalized Cohen--Macaulay modules with maximal depth are classified. Finally,
the attached primes of $H^i_{\mathfrak{m}}(M)$ are considered for
$i<\mathrm{dim} M$.
