Translational tiling is a covering of a space (such as Euclidean space) using translated copies of one building block, called a "translational tile'', without any positive measure overlaps.     Can we determine whether a given set is a translational tile?  Does any translational tile admit a periodic tiling?  A well known argument shows that these two questions are closely related.  In the talk, we will discuss this relation and present some new developments, joint with Terence Tao, establishing answers to both questions.