  A weighted pointed curve consists of a nodal curve and a sequence of marked
smooth points, each assigned a number between zero and one. A subset of the
marked points may coincide if the sum of the corresponding weights is no
greater than one. We construct moduli spaces for these objects using methods of
the log minimal model program, and describe the induced birational morphisms
between moduli spaces as the weights are varied. In the genus zero case, we
explain the connection to Geometric Invariant Theory quotients of points in the
projective line, and to compactifications of moduli spaces studied by Kapranov,
Keel, and Losev-Manin.
