  We consider the metrics for cosmic strings and $p$-branes in spacetime
dimension $N>4$, that is, we look for solutions to Einstein-Maxwell-Dilaton
gravity in $N$-dimensions with boost symmetry in the $p$-directions along the
brane. Focussing first in detail on the five dimensional uncharged cosmic
string we discuss the solution, which turns out to have a naked singularity on
the brane, as well as considering its Kaluza-Klein reduction. We show how
singularities may be avoided with particular core models. We then derive the
general uncharged $p$-brane solution in arbitrary dimension. Finally, we
consider an Einstein-Maxwell-Dilaton action, with arbitrary value of the
dilaton coupling parameter, deriving the solutions for electrically and
magnetically charged branes, as well as a class of self-dual branes.
