The goal of the presented work is the efficient computation of Maxwell boundary and eigenvalue problems using high order H(curl) finite elements. We discuss a systematic strategy for the realization of arbitrary order hierarchic H(curl)conforming finite elements for triangular and tetrahedral element geometries. The shape functions are classified as lowestorder Nédélec, higher-order edge-based, face-based (only in 3D) and element-based ones. Our new shape functions provide not only the global complete sequence property, but also local complete sequence properties for each edge-, face-, element-block. This local property allows an arbitrary variable choice of the polynomial degree for each edge, face, and element. A second advantage of this construction is that simple block-diagonal preconditioning gets efficient. Our high order shape functions contain gradient shape functions explicitly. In the case of a magnetostatic boundary value problem, the gradient basis functions can be skipped, which reduces the problem size, and improves the condition number. We successfully apply the new high order elements for a 3D magnetostatic boundary value problem, and a Maxwell eigenvalue problem showing severe edge and corner singularities.