  We classify all unitary modular tensor categories (UMTCs) of rank $\leq 4$.
There are a total of 70 UMTCs of rank $\leq 4$ (Note that some authors would
have counted as 35 MTCs.) In our convention there are two trivial unitary MTCs
distinguished by the modular $S$ matrix $S=(\pm1)$. Each such UMTC can be
obtained from 10 non-trivial prime UMTCs by direct product, and some symmetry
operations.
  UMTCs encode topological properties of anyonic quantum systems and can be
used to build fault-tolerant quantum computers. We conjecture that there are
only finitely many equivalence classes of MTCs for any given rank, and a UMTC
is universal for anyonic quantum computation if and only if its global quantum
dimension $D^2$ is \emph{not} an integer.
  Discovery of non-abelain anyons in Nature will be a landmark in condensed
matter physics. The non-abelian anyons in UMTCs of rank $\leq 4$ are the
simplest, and, therefore, are most likely to be found. G. Moore and N. Read
proposed that non-abelian statistics could occur in fractional quantum Hall
(FQH) liquids. The Read-Rezayi conjecture predicts the existence of anyons
related to $SU(2)_k$ in FQH liquids at filling fractions $\nu=2+\frac{k}{k+2}$
for $k=1,2,3$. For $\nu={5/2}$ and $SU(2)_2$, there is a numerical proof and
experimental evidence for this conjecture. The Ising anyons might exist in
chiral superconductors (strontium ruthenate). There are theoretical designs for
the toric code MTC using Josephson junction array, for the $Fib\times Fib$ MTC
using optical lattice, and for the $SU(2)_k$ using cold trapped bosonic atoms.
