  For vector-valued Maass cusp forms for~$SL_2(\mathbb{Z})$ with real
weight~$k\in\mathbb{R}$ and spectral parameter $s\in\mathbb{C}$, $\mathrm{Re}
s\in (0,1)$, $s\not\equiv \pm k/2$ mod $1$, we propose a notion of
vector-valued period functions, and we establish a linear isomorphism between
the spaces of Maass cusp forms and period functions by means of a cohomological
approach. The period functions are a generalization of those for the classical
Maass cusp forms, being solutions of a finite-term functional equation or,
equivalently, eigenfunctions with eigenvalue $1$ of a transfer operator deduced
from the geodesic flow on the modular surface. We apply this result to deduce a
notion of period functions and related linear isomorphism for Jacobi Maass
forms of weight $k+1/2$ for the semi-direct product of $SL_2(\mathbb{Z})$ with
the integer points $Hei(\mathbb{Z})$ of the Heisenberg group.
