  We propose a formalism for tilings with infinite local complexity (ILC), and
especially fusion tilings with ILC. We allow an infinite variety of tile types
but require that the space of possible tile types be compact. Examples include
solenoids, pinwheel tilings, tilings with fault lines, and tilings with
infinitely many tile sizes, shapes, or labels. Special attention is given to
tilings where the infinite local complexity comes purely from geometry (shears)
or comes purely from combinatorics (labels). We examine spectral properties of
the invariant measures and define a new notion of complexity that applies to
ILC tilings.
