  For an element $\Psi$ in the graded vector space $\Omega^*(M, TM)$ of tangent
bundle valued forms on a smooth manifold $M$, a $\Psi$-submanifold is defined
as a submanifold $N$ of $M$ such that $\Psi_{|N} \in \Omega^*(N, TN)$. The
class of $\Psi$-submanifolds encompasses calibrated submanifolds, complex
submanifolds and all Lie subgroups in compact Lie groups. The graded vector
space $\Omega^*(M, TM)$ carries a natural graded Lie algebra structure, given
by the Fr\"olicher-Nijenhuis bracket $[-,- ]^{FN}$. When $\Psi$ is an odd
degree element with $[ \Psi, \Psi]^{FN} =0$, we associate to a
$\Psi$-submanifold $N$ a strongly homotopy Lie algebra, which governs the
formal and (under certain assumptions) smooth deformations of $N$ as a
$\Psi$-submanifold, and we show that under certain assumptions these
deformations form an analytic variety. As an application we revisit formal and
smooth deformation theory of complex closed submanifolds and of
$\varphi$-calibrated closed submanifolds, where $\varphi$ is a parallel form in
a real analytic Riemannian manifold.
