  We introduce a technique for recovering a sufficiently smooth function from
its ray transform over a wide class of curves in a general region of Euclidean
space. The method is based on a complexification of the underlying vector
fields defining the initial transport and recasting the problem in terms of
complex-analytic function theory. Explicit inversion formulae are then given in
a unified form. The method is then used to give inversion formulae for the
attenuated ray transform.
