Answer
All real numbers except $-2$ and $2$.
Work Step by Step
We know that $x^2-4=(x+2)(x-2).$
The denominator cannot be $0$, hence $x+2\ne0$ and $x-2\ne0$, therefore $x\ne-2$ and $x\ne2$.
Thus the domain is all real numbers except $-2$ and $2$.