  For several applications in the arithmetic of abelian varieties it is
important to compute canonical heights. Following Faltings and Hriljac, we show
how the canonical height on the Jacobian of a smooth projective curve can be
computed using arithmetic intersection theory on a regular model of the curve
in practice. In the case of hyperelliptic curves we present a complete
algorithm that has been implemented in Magma. Several examples are computed and
the behavior of the running time is discussed.
