  There are two motivations to consider statistics that are neither Bose nor
Fermi: (1) to extend the framework of quantum theory and of quantum field
theory, and (2) to provide a quantitative measure of possible violations of
statistics. After reviewing tests of statistics for various particles, and
types of statistics that are neither Bose nor Fermi, I discuss quons, particles
characterized by the parameter $q$, which permit a smooth interpolation between
Bose and Fermi statistics; $q=1$ gives bosons, $q=-1$ gives fermions. The new
result of this talk is work by Robert C. Hilborn and myself that gives a
heuristic argument for an extension of conservation of statistics to quons with
trilinear couplings of the form $\bar{f}fb$, where $f$ is fermion-like and $b$
is boson-like. We showed that $q_f^2=q_b$. In particular, we related the bound
on $q_{\gamma}$ for photons to the bound on $q_e$ for electrons, allowing the
very precise bound for electrons to be carried over to photons. An extension of
our argument suggests that all particles are fermions or bosons to high
precision.
