Forcing with adequate sets of models as side conditions

We present a general framework for forcing on (2) with finite conditions using countable models as side conditions. This framework is based on a method of comparing countable models as being membership related up to a large initial segment. We give several examples of this type of forcing, including adding a function on (2), adding a nonreflecting stationary subset of 2 cof (), and adding an (1)-Kurepa tree.