  We review the recently developed supersymmetric extensions of field theories
with non-standard kinetic terms (so-called K field theories) in two an three
dimensions. Further, we study the issue of topological defect formation in
these supersymmetric theories. Specifically, we find supersymmetric K field
theories which support topological kinks in 1+1 dimensions as well as
supersymmetric extensions of the baby Skyrme model for arbitrary nonnegative
potentials in 2+1 dimensions.
