How many positive multiples of $9$ are less than $1000$?
First, we'll find the largest multiple of $9$ less than $1000$.  Because $1000 \div 9 = 111$ remainder 1, the largest multiple of $9$ less than $1000$ is $111 \cdot 9$, or $999$.

Therefore, every number in the form $n \cdot 9$ works, as long as $n$ is positive and at most $111$.  There are $111$ such values of $n$, so there are $\boxed{111}$ multiples of $9$ that are less than $1000$.