  We present new algorithms for computing adjoint ideals of curves and thus, in
the planar case, adjoint curves. With regard to terminology, we follow
Gorenstein who states the adjoint condition in terms of conductors. Our main
algorithm yields the Gorenstein adjoint ideal G of a given curve as the
intersection of what we call local Gorenstein adjoint ideals. Since the
respective local computations do not depend on each other, our approach is
inherently parallel. Over the rationals, further parallelization is achieved by
a modular version of the algorithm which first computes a number of the
characteristic p counterparts of G and then lifts these to characteristic zero.
As a key ingredient, we establish an efficient criterion to verify the
correctness of the lift. Well-known applications are the computation of
Riemann-Roch spaces, the construction of points in moduli spaces, and the
parametrization of rational curves. We have implemented different variants of
our algorithms together with Mnuk's approach in the computer algebra system
Singular and give timings to compare the performance of the algorithms.
