  It is a well-known result of Etingof, Nikshych and Ostrik that there are
finitely many inequivalent integral modular categories of any fixed rank $n$.
This follows from a double-exponential bound on the maximal denominator in an
Egyptian fraction representation of 1. A na\"ive computer search approach to
the classification of rank $n$ integral modular categories using this bound
quickly overwhelms the computer's memory (for $n\geq 7$). We use a modified
strategy: find general conditions on modular categories that imply integrality
and study the classification problem in these limited settings. The first such
condition is that the order of the twist matrix is 2,3,4 or 6 and we obtain a
fairly complete description of these classes of modular categories. The second
condition is that the unit object is the only simple non-self-dual object,
which is equivalent to odd-dimensionality. In this case we obtain a (linear)
improvement on the bounds and employ number-theoretic techniques to obtain a
classification for rank at most 11 for odd-dimensional modular categories.
