  We address the Riemann and Cauchy problems for systems of $n$ conservation
laws in $m$ unknowns which are subject to $m-n$ constraints ($m\geq n$). Such
constrained systems generalize systems of conservation laws in standard form to
include various examples of conservation laws in Physics and Engineering beyond
gas dynamics, e.g., multi-phase flow in porous media. We prove local
well-posedness of the Riemann problem and global existence of the Cauchy
problem for initial data with sufficiently small total variation, in one
spatial dimension. The key to our existence theory is to generalize the
$m\times n$ systems of constrained conservation laws to $n\times n$ systems of
conservation laws with states taking values in an $n$-dimensional manifold and
to extend Lax's theory for local existence as well as Glimm's random choice
method to our geometric framework. Our resulting existence theory allows for
the accumulation function to be non-invertible across hypersurfaces.
