  If a function $f:\mathbb{R}\to\mathbb{R}$ can be represented as the sum of
$n$ periodic functions as $f=f_1+\dots+f_n$ with $f(x+\alpha_j)=f(x)$
($j=1,\dots,n$), then it also satisfies a corresponding $n$-order difference
equation $\Delta_{\alpha_1}\dots\Delta_{\alpha_n} f=0$. The periodic
decomposition problem asks for the converse implication, which may hold or fail
depending on the context (on the system of periods, on the function class in
which the problem is considered, etc.). The problem has natural extensions and
ramifications in various directions, and is related to several other problems
in real analysis, Fourier and functional analysis. We give a survey about the
available methods and results, and present a number of intriguing open
problems.
