Tiling by translates of a function

A function f on the real line is said to tile by translates
along a discrete set $\Lambda$ if the sum of all the functions
f(x-\lambda), $\lambda \in \Lambda$, is equal to one identically.
Which functions can tile by translates, and what can be said
about the translation set $\Lambda$? I will survey the subject and
discuss some recent results joint with Mihail Kolountzakis.