At least as a counter-point to other reasonable viewpoints: I am not such an enthusiast about measure theory as "foundation" for integrals, because it seems to me that measure theory over-emphasizes just one smallish part of what we really want. Namely, starting with (ambiguously) nice functions $f_n$, perhaps continuous and compactly supported on a Euclidean space, so that any notion of integral makes sense for the individuals, _one_thing_ we want is that $\lim_n \int f_n=\int\lim_n f_n$. Part of the issue is in what sense the limit of the functions themselves is taken, and there is not a unique answer to this. But it is clear from examples that pointwise limits of continuous functions may be harder to sensibly integrate in naive terms.
Another fundamental issue is the distinction between integrating differential forms and integrating functions, although again I claim this is more of a technical distinction than fundamental. Namely, while from a "measure-theory" viewpoint integrating $f$ on an interval $[a,b]$ does not even mention direction, in usual calculus, and if we imagine we are integrating the differential form $f\,dx$, direction does matter. (That is, in elementary calculus the integral $\int_a^b$ is the negative of $\int_b^a$.) This issue is still more notational than substantive even when doing path integrals and surface integrals. It seems to me that the issue is serious only when we are doing calculus on manifolds, where there is no canonical coordinate system, so that a differential form has no canonical local expression, and the only thing that is well-defined is an integral of an exterior $k$-form over a $k$-cycle or $k$-current, etc.
Still, in all cases, the external properties and desiderata are very similar, so that all these things are more the same than different.
Even when (as in basic Lebesgue theory) we discover that some restrictions are necessary to be able to pass a linear operator inside an integral, it seems to me that the development of mathematics has been to make sense of otherwise-dubious outcomes, whenever possible. For example, $\int_{\mathbb R} {e^{ix\xi}\;dx\over 1+x^2}$ cannot be differentiated with respect to $\xi$ very much without producing a not-classically-sensible integral. But as Fourier transform of a tempered distribution it's still completely fine.
