In this talk, we consider fractional differential equations posed on the full space R^d. A distinct advantage of full-space formulations for fractional PDEs is that all common definitions of non-integer powers of differential operators are equivalent, which is not true for formulations on bounded domains. In order to treat the full-space problem numerically, serveral reformulations have to be made. Starting with the well-known Caffarelli-Silvestre extension to R^d×R⁺, we truncate the extension problem in the extended variable only to R^d × (0,Y) for some Y > 0. Then, a diagonalization procedure, similar to the case of the spectral fractional Laplacian on bounded domains, can be employed that leads to a sequence of scalar Helmholtztype problems, which are discretized with a symmetric coupling of finite elements and boundary elements. Combined with a hp-FEM discretization in the extended variable, this gives a fully computable approximation with reasonable computational effort. For the mentioned reformulations, we show well-posedness in certain exotic Hilbert spaces. Using purely variational techniques, we derive an algebraic rate of decay of the solution of the truncated problem to the full-space solution as Y → ∞ as well as estimates of weighted analytic type for higher order derivatives of the truncated extension problem. These decay and regularity estimates can be used to derive a priori estimates for the error between the exact full-space solution and an approximation based on our approach using a coupling of finite elements and boundary elements.