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We consider walks on the edges of the square lattice $\mathbb Z^2$ which obey\emph{two-step rules,} which allow (or forbid) steps in a given direction to befollowed by steps in another direction. We classify these rules according to anumber of criteria, and show how these properties affect their generatingfunctions, asymptotic enumerations and limiting shapes, on the full lattice aswell as the upper half plane.  For walks in the quarter plane, we only make a few tentative first steps. Wepropose candidates for the group of a model, analogous to the group of aregular short-step quarter plane model, and investigate which models havefinite versus infinite groups. We demonstrate that the orbit sum method used tosolve a number of the original models can be made to work for some models here,producing a D-finite solution. We also generate short series for all models andguess differential or algebraic equations where possible. In doing so, we findthat there are possibilities here which do not occur for the regular short-stepmodels, including cases with algebraic or D-finite generating functions butinfinite groups, as well as models with non-D-finite generating functions butfinite groups.

Walks obeying two-step rules on the square lattice: full, half and quarter planes