Is there an analogue of uniform continuity for multivariable functions? And, is addition and multiplication uniformly continuous?

Question

I know the definition of uniform continuity for single variable functions from the reals to the reals. Is there an analogue of that definition for multivariable functions from $\mathbb{R}^n$ to $\mathbb{R}$, where $n \geq 2$? And with that definition, is it possible to prove that the addition function $f(x,y)=x+y$ is uniformly continuous over all of $\mathbb{R}^2$, whereas the multiplication function $g(x,y)=x \times y$ is not uniformly continuous over all of $\mathbb{R}^2$?