TITLE: Why differential forms are important?
QUESTION [50 upvotes]: Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated without them: for example if you want to learn the definition of symplectic manifold, you must first learn what is differential form. 
However I have heard another opinion from people with non-geometric background (say, functional analysis, probability) that differential forms are probably used only in problems where they already appear in basic definitions (like symplectic geometry). When I taught an undergraduate course "Analysis on manifold" I had a feeling that my students might have got the same impression at the end: in the second half of the semester I developed calculus of differential forms with rather few concrete applications.  Personally I am completely convinced in usefulness of differential forms, but find it not so easy to explain this to people who are not used to them.
I would like to make a list of concrete applications of differential forms. Also it would be interesting to know historical reasons to introduce them. To start the list, below are some examples which come to my mind. 
1) The general Stokes formula $\int_M d\omega=\pm \int_{\partial M}\omega$ generalizes all classical formulas of Green, Gauss, Stokes.
2) The Haar measure on a Lie group can be easily constructed as a translation invariant top differential form. (Some experts often prefer to consider this as a special case of a more general and more difficult fact of existence of Haar measure on locally compact groups. In that generality there is no such a simple and relatively explicit description of Haar measure as there is no language similar to differential forms.)
3) The cohomology of the de Rham complex of differential forms on a manifold is canonically isomorphic to singular cohomology (with real coefficients) of the manifold. One use of this isomorphism is that even Betti numbers of a symplectic manifold are non-zero. Another non-trivial use of this fact is the Hodge decomposition of the cohomology (with complex coefficients) of a compact Kahler manifold which makes it, in particular, a bi-graded algebra (rather than just graded) and provides new information on the Betti numbers, say $\beta_{2k+1}$ must be even.

REPLY [4 votes]: There is at least one more important example where people who believe that "differential forms are used only in problems where they already appear in basic definitions" are completely wrong. Differential forms and the exterior differential were created (or discovered) by Elie Cartan in the 1890's in order to solve the Pfaff problem, as follows.
Let us think in $R^n$. Certainly these people will admit that a differential form of degree one, that is, a covector field $$\omega=f_1dx_1+...+f_ndx_n$$ is a natural object and deserves some attention... Here $f_1, \dots, f_n$ are real functions of $x_1, \dots, x_n$. A very natural and universal method in sciences, in order to study an object, is to consider it in the coordinates system where it gets the simplest form... Pfaff's problem is: given such a form $\omega$, in a small neighborhood of a given point, can I reduce the number of variables to some $k<n$? In other words, can I find local curvilinear coordinates $y_1, \dots, y_n$ such that $$\omega=g_1dy_1+...+g_kdy_k$$ where $g_1, \dots, g_k$ are real functions of $y_1, \dots, y_k$ ?
In order to solve this, Cartan defines the exterior differential $2$-form $d\omega$, and considers what he calls the successive differentials of $\omega$, namely:
$\omega'=d\omega$,
$\omega"=\omega\wedge d\omega$,
$\omega'''=d\omega\wedge d\omega$, $\dots$
Generally, he defines $\omega^{(k)}$ as $(d\omega)^{(k+1)/2}$ (using the wedge product) if $k$ is odd, and as $\omega\wedge(d\omega)^{k/2}$ if $k$ is even. He proves that the exterior differential $d$ is invariant under coordinates changes (quite a discovery!) and concludes that a necessary condition for the $k$-reducibility is that $\omega^{(\ell)}$ vanishes for every $l>k$. Then, his real work is to prove that this necessary condition is sufficient, in the real-analytic case (the smooth case requires some more subtle nondegeneracy condition, if I'm correct).
Also note how these simple formulas contain a lot of geometry. Foliations, symplectic and contact geometries, appear here in the consideration of the (non)vanishing of the successive derivates. Differential forms were indeed one of Elie Cartan's great discoveries.