Let $f(x)$ and $g(x)$ be odd functions.  Is $f(x) g(x)$ even, odd, or neither?

Enter "odd", "even", or "neither".
Since $f(x)$ and $g(x)$ are odd functions,
\[f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x),\]so $f(x) g(x)$ is an $\boxed{\text{even}}$ function.