Does continuity guarantee existence of double integral?

In single variable real analysis we know if $f: mathbb{R} to mathbb{R}$ is continuous then its integral exists over any bounded domain. Does this generalize to the 2-variable case? Does the double integral of a function $f: mathbb{R^2} to mathbb{R}$ exist over any bounded domain when the function is continuous, or is uniform continuity (or some other stronger condition) required?