  We investigate the QCD chiral phase transition at finite chemical potential
$\mu$, using the renormalization group (RG) to characterize the infrared
behavior of sigma models constrained by the flavor and spacetime symmetries.
The results are similar to those obtained from RG analysis of the finite
temperature transition at zero baryon density. When there are more than two
massless flavors of quarks, a first order transition is predicted for the
entire phase boundary. In the two flavor case, a boundary with first and second
order regions separated by a tricritical point seems most likely. We discuss
the real-world case with two light quarks and an intermediate mass strange
quark. Improved lattice data on the temperature transition will strongly
constrain the possibilities for the phase boundary.
