Abstract: A set K tiles Rn by translations if {K + k}{k in Zn} is a partition of Rn . I will discuss the interplay between these sets and n by n matrices. In particular, I will discuss the following three questions:(1) For which matrices A do there exist sets K that tile by translation and such that K is a subset of A(K)?(2) For which matrices A do there exist sets K that tile by translation such that the closure of K is contained in A(K)?(3) For which matrices A do there exist sets K that tile by translation and {Aj(K)}{j in Z} is a partition of Rn ?Many examples will be given of sets as in (1,2,3) above satisfying additional symmetry/convexity conditions. Any proofs given will be restricted to the 1 or 2 dimensional case. Open problems will also be given.