  We review recent advances in the spectral theory of Schr\"odinger operators
with decaying potentials. The area has seen spectacular progress in the past
few years, stimulated by several conjectures stated by Barry Simon starting at
the 1994 International Congress on Mathematical Physics in Paris. The
one-dimensional picture is now fairly complete, and provides many striking
spectral examples. The multidimensional picture is still far from clear and may
require deep original ideas for further progress. It might hold the keys for
better understanding of a wide range of spectral and dynamical phenomena for
Schr\"odinger operators in higher dimensions.
