We give a rigorous formulation of the planar stochastic heat
equation, whose solution is the free energy of an undirected random
polymer, using a version of the Skorokhod integral. The solution is
represented as an $L^1$ limit of a martingale given by the Feynman-Kac
formula. We also show that the fundamental solution far from the center
has fluctuations given by the 1+1 KPZ equation. This is a joint work
with Jeremy Quastel and Balint Virag.