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Abstract: We analyze higher order finite elements applied to second order elliptic interface problems. Our a priori error estimates in the - and -norm are expressed in terms of the approximation order p and a parameter δ that quantifies how well the interface is resolved by the finite element mesh. The optimal p-th order convergence in the -norm is only achieved under stringent assumptions on δ, namely, . Under weaker conditions on δ, optimal a priori estimates can be established in the - and in the -norm, where is a subdomain that excludes a tubular neighborhood of the interface of width . In particular, if the interface is approximated by an interpolation spline of order p and if full regularity is assumed, then optimal convergence orders and p for the approximation in the - and the -norm can be expected but not order p for the approximation in the -norm. Numerical examples in 2D and 3D illustrate and confirm our theoretical results. [Copyright &y& Elsevier]