  Loss of compactness that occurs in may significant PDE settings can be
expressed in a well-structured form of profile decomposition for sequences.
Profile decompositions are formulated in relation to a triplet $(X,Y,D)$, where
$X$ and $Y$ are Banach spaces, $X\hookrightarrow Y$, and $D$ is, typically, a
set of surjective isometries on both $X$ and $Y$. A profile decomposition is a
representation of a bounded sequence in $X$ as a sum of elementary
concentrations of the form $g_kw$, $g_k\in D$, $w\in X$, and a remainder that
vanishes in $Y$. A necessary requirement for $Y$ is, therefore, that any
sequence in $X$ that develops no $D$-concentrations has a subsequence
convergent in the norm of $Y$. An imbedding $X\hookrightarrow Y$ with this
property is called $D$-cocompact, a property weaker than, but related to,
compactness. We survey known cocompact imbeddings and their role in profile
decompositions.
