  In this paper we consider a class of unfitted finite element methods for
discretization of partial differential equations on surfaces. In this class of
methods known as the Trace Finite Element Method (TraceFEM), restrictions or
traces of background surface-independent finite element functions are used to
approximate the solution of a PDE on a surface. We treat equations on steady
and time-dependent (evolving) surfaces. Higher order TraceFEM is explained in
detail. We review the error analysis and algebraic properties of the method.
The paper navigates through the known variants of the TraceFEM and the
literature on the subject.
