We study solution techniques for problems involving fractional powers of symmetric, coercive and elliptic operators. These can be realized as the Dirichlet to Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder, which we analyze in the framework of Muckenhoupt weighted Sobolev spaces. Motivated by the rapid decay of the solution to this problem, we propose a truncation that is suitable for numerical approximation. We discretize this truncation using first degree tensor product finite elements. We derive suboptimal a priori error estimates for quasi-uniform discretizations and quasi-optimal error estimates for anisotropic discretizations. We explore extensions and applications of the a priori theory previously described: a posteriori error analysis and adaptivity; parabolic equations with fractional diffusion and Caputo fractional time derivative; elliptic and parabolic fractional obstacle problems; and optimal control problems.