  We study the phases and fixed-point structure of two-dimensional
supersymmetric Wess-Zumino models with one supercharge. Our work is based on
the functional renormalization group formulated in terms of a manifestly
off-shell supersymmetric flow equation for the effective action. Within the
derivative expansion, we solve the flow of the superpotential also including
the anomalous dimension of the superfield. The models exhibit a surprisingly
rich fixed-point structure with a discrete number of fixed-point
superpotentials. Each fixed-point superpotential is characterized by its number
of nodes and by the number of RG relevant directions. In limiting cases, we
find periodic superpotentials and potentials which confine the fields to a
compact target space. The maximally IR-attractive fixed point has one relevant
direction, the tuning of which distinguishes between supersymmetric and broken
phases. For the Wess-Zumino model defined near the Gaussian fixed point, we
determine the phase diagram and compute the corresponding ground-state masses.
