  Partial differential equations posed on surfaces arise in a number of
applications. In this survey we describe three popular finite element methods
for approximating solutions to the Laplace-Beltrami problem posed on an
$n$-dimensional surface $\gamma$ embedded in $\mathbb{R}^{n+1}$: the
parametric, trace, and narrow band methods. The parametric method entails
constructing an approximating polyhedral surface $\Gamma$ whose faces comprise
the finite element triangulation. The finite element method is then posed over
the approximate surface $\Gamma$ in a manner very similar to standard FEM on
Euclidean domains. In the trace method it is assumed that the given surface
$\gamma$ is embedded in an $n+1$-dimensional domain $\Omega$ which has itself
been triangulated. An $n$-dimensional approximate surface $\Gamma$ is then
constructed roughly speaking by interpolating $\gamma$ over the triangulation
of $\Omega$, and the finite element space over $\Gamma$ consists of the trace
(restriction) of a standard finite element space on $\Omega$ to $\Gamma$. In
the narrow band method the PDE posed on the surface is extended to a
triangulated $n+1$-dimensional band about $\gamma$ whose width is proportional
to the diameter of elements in the triangulation. In all cases we provide
optimal a priori error estimates for the lowest-order finite element methods,
and we also present a posteriori error estimates for the parametric and trace
methods. Our presentation focuses especially on the relationship between the
regularity of the surface $\gamma$, which is never assumed better than of class
$C^2$, the manner in which $\gamma$ is represented in theory and practice, and
the properties of the resulting methods.
