We define higher-order analogs to the piecewise linear surface finite element method studied in \cite{Dz88} and prove error estimates in both pointwise and $L_2$-based norms.   Using the Laplace-Beltrami problem on an implicitly defined surface $\Gamma$ as a model PDE, we define Lagrange finite element methods of arbitrary degree on polynomial approximations to $\Gamma$ which likewise are of arbitrary degree.  Then we prove a priori error estimates in the $L_2$, $H^1$, and corresponding pointwise norms that demonstrate the interaction between the ``PDE error'' that arises from employing a finite-dimensional finite element space and the ``geometric error'' that results from approximating $\Gamma$.  We also consider parametric finite element approximations that are defined on $\Gamma$ and thus induce no geometric error.  Computational examples confirm the sharpness of our error estimates.