  We develop an invariant deformation theory, in a form accessible to practice,
for affine schemes $W$ equipped with an action of a reductive algebraic group
$G$. Given the defining equations of a $G$-invariant subscheme $X \subset W$,
we device an algorithm to compute the universal deformation of $X$ in terms of
generators and relations up to a given order. In many situations, our algorithm
even computes an algebraization of the universal deformation. As an
application, we determine new families of examples of the invariant Hilbert
scheme of Alexeev and Brion, where $G$ is a classical group acting on a
classical representation, and describe their singularities.
