  We develop a general theory of transport-limited aggregation phenomena
occurring on curved surfaces, based on stochastic iterated conformal maps and
conformal projections to the complex plane. To illustrate the theory, we use
stereographic projections to simulate diffusion-limited-aggregation (DLA) on
surfaces of constant Gaussian curvature, including the sphere ($K>0$) and
pseudo-sphere ($K<0$), which approximate "bumps" and "saddles" in smooth
surfaces, respectively. Although curvature affects the global morphology of the
aggregates, the fractal dimension (in the curved metric) is remarkably
insensitive to curvature, as long as the particle size is much smaller than the
radius of curvature. We conjecture that all aggregates grown by conformally
invariant transport on curved surfaces have the same fractal dimension as DLA
in the plane. Our simulations suggest, however, that the multifractal
dimensions increase from hyperbolic ($K<0$) to elliptic ($K>0$) geometry, which
we attribute to curvature-dependent screening of tip branching.
