  We study dark matter production by decaying topological defects, in
particular cosmic strings. In topological defect or "top-down" (TD) scenarios,
the dark matter injection rate varies as a power law with time with exponent
$p-4$. We find a formula in closed form for the yield for all $p < 3/2$, which
accurately reproduces the solution of the Boltzmann equation. We investigate
two scenarios ($p=1$, $p=7/6$) motivated by cosmic strings which decay into
TeV-scale states with a high branching fraction into dark matter particles. For
dark matter models annihilating either by s-wave or p-wave, we find the regions
of parameter space where the TD model can account for the dark matter relic
density as measured by Planck. We find that topological defects can be the
principal source of dark matter, even when the standard freeze-out calculation
under-predicts the relic density and hence can lead to potentially large "boost
factor" enhancements in the dark matter annihilation rate. We examine dark
matter model-independent limits on this scenario arising from unitarity and
discuss example model-dependent limits coming from indirect dark matter search
experiments. In the four cases studied, the upper bound on $G\mu$ for strings
with an appreciable channel into TeV-scale states is significantly more
stringent than the current Cosmic Microwave Background limits.
