  It is well known that every modular form~$f$ on a discrete subgroup
$\Gamma\leqslant \textrm{SL}(2, \mathbb R)$ satisfies a third-order nonlinear
ODE that expresses algebraic dependence of the functions~$f$, $f'$, $f''$
and~$f'''$. These ODEs are automatically invariant under the Lie group
$\textrm{SL}(2, \mathbb R)$, which acts on the solution spaces thereof with an
open orbit (and the discrete stabiliser~$\Gamma$ of a generic solution).
Similarly, every modular form satisfies a fourth-order nonlinear ODE that is
invariant under the Lie group $\textrm{GL}(2, \mathbb R)$ acting on its
solution space with an open orbit. ODEs for modular forms can be compactly
expressed in terms of the differential invariants of these actions. The
invariant forms of both ODEs define plane algebraic curves naturally associated
with every modular form; the corresponding ODEs can be seen as modular
parametrisations of the associated curves.
  After reviewing examples of nonlinear ODEs satisfied by classical modular
forms (such as Eisenstein series, modular forms on congruence subgroups of
level two and three, theta constants, and some newforms of weight two), we
generalise these results to Jacobi forms; these satisfy involutive third-order
PDE systems that are invariant under the Lie group $\textrm{SL}(2, \mathbb
R)\ltimes H$ where $H$ is the Heisenberg group.
