This thesis is an extension (and generalization) of the work(s) done by Varayu Boonpogkrong and Jaroslav Kurzweil. Stokes’s theorem, the generalization of the fundamental theorem of calculus, is all about comparing integrals over manifolds and integrals over their boundaries. Stokes’ theorem on manifolds says that the integral of a differential k-form over the boundary of a compact oriented differentiable manifold M is equal to the integral of the exterior derivative of that form over M. Our approach to integration over a general manifold has several distinguishing features: the manifold M must be oriented and the differential form must have compact support. Note that on a manifold of dimension n, one can integrate only n-forms, not functions. Furthermore, the boundary of M has the boundary orientation induced from M. To orient a manifold M, we orient the tangent space at each point p in M . This can be done by simply assigning a nonzero n-covector to each point of M, in other words, by giving a nowhere-vanishing n-form on M. The assignment of an orientation at each point must be done in a “coherent’’ way, so that the orientation does not change abruptly in a neighborhood of a point. The simplest way to guarantee this is to require that the n-form on M specifying the orientation at each point be smooth . It is proved that a manifold M of dimension n has a smooth nowhere-vanishing n-form if and only if it has an oriented atlas. It should be recalled that an orientation on a manifold M with boundary induces, in a natural way, an orientation on the boundary of M . The prototype of a manifold with boundary is the closed upper half-space H n with the suace topology of R n . We use H n to serve as a model for manifolds with boundary. A topological n-manifold with boundary is a second countable Hausdorff topological space which is locally H n . A smooth manifold with boundary is a topological manifold M with boundary together with a maximal smooth atlas. The definition of the boundary orientation is independent of the oriented atlas for M. One new way to prove the Stokes’ theorem on manifolds is the Kurzweil-Henstock approach. Here we use the physical definition of an exterior derivative and k-forms to prove Stokes’ theorem by the Kurzweil-Henstock approach. The partition of unity plays an important role in the integral on manifolds. Kurzweil and Jarnik defined the Kurzweil-Henstock integral using the partition of unity. The Kurzweil-Henstock integral using partition of unity is equivalent to the Lebesgue integral in the n-dimensional Euclidean space. This integral can be used for integration on manifolds without details. The on manifolds is defined using change of variables formula. The Kurzweil- Henstock integral is defined by Riemann sums. In fact it’s an integral of Riemann type. Here we define the exterior derivative d ? of an elementary n-form ? by the means of parallelograms. In other words the exterior derivative d ? is a point-parallelogram function. The value of the integral does not depend on charts. Hence the integral value is uniquely determined.