  We say that a function $f \in L^1(\mathbb{R})$ tiles at level $w$ by a
discrete translation set $\Lambda \subset \mathbb{R}$, if we have
$\sum_{\lambda \in \Lambda} f(x-\lambda)=w$ a.e. In this paper we survey the
main results, and prove several new ones, on the structure of tilings of
$\mathbb{R}$ by translates of a function. The phenomena discussed include
tilings of bounded and of unbounded density, uniform distribution of the
translates, periodic and non-periodic tilings, and tilings at level zero.
Fourier analysis plays an important role in the proofs. Some open problems are
also given.
