  We develop a non-singular theory of three-dimensional dislocation loops in a
particular version of Mindlin's anisotropic gradient elasticity with up to six
length scale parameters. The theory is systematically developed as a
generalization of the classical anisotropic theory in the framework of
linearized incompatible elasticity. The non-singular version of all key
equations of anisotropic dislocation theory are derived as line integrals,
including the Burgers displacement equation with isolated solid angle, the
Peach-Koehler stress equation, the Mura-Willis equation for the elastic
distortion, and the Peach-Koehler force. The expression for the interaction
energy between two dislocation loops as a double line integral is obtained
directly, without the use of a stress function. It is shown that all the
elastic fields are non-singular, and that they converge to their classical
counterparts a few characteristic lengths away from the dislocation core. In
practice, the non-singular fields can be obtained from the classical ones by
replacing the classical (singular) anisotropic Green's tensor with the
non-singular anisotropic Green's tensor derived by \cite{Lazar:2015ja}. The
elastic solution is valid for arbitrary anisotropic media. In addition to the
classical anisotropic elastic constants, the non-singular Green's tensor
depends on a second order symmetric tensor of length scale parameters modeling
a weak non-locality, whose structure depends on the specific class of crystal
symmetry. The anisotropic Helmholtz operator defined by such tensor admits a
Green's function which is used as the spreading function for the Burgers vector
density. As a consequence, the Burgers vector density spreads differently in
different crystal structures.
