  We develop a unified theory for continuous in time finite element
discretisations of partial differential equations posed in evolving domains
including the consideration of equations posed on evolving surfaces and bulk
domains as well coupled surface bulk systems. We use an abstract variational
setting with time dependent function spaces and abstract time dependent finite
element spaces. Optimal a priori bounds are shown under usual assumptions on
perturbations of bilinear forms and approximation properties of the abstract
finite element spaces. The abstract theory is applied to evolving finite
elements in both flat and curved spaces. Evolving bulk and surface
isoparametric finite element spaces defined on evolving triangulations are
defined and developed. These spaces are used to define approximations to
parabolic equations in general domains for which the abstract theory is shown
to apply. Numerical experiments are described which confirm the rates of
convergence.
