The thesis is mainly concerned with two concepts fundamental for microlocal analysis, namely the wave front set and oscillatory integrals. Many definitions and results are generalized to manifolds and vector bundles, and for this reason the generalization of classical distribution theory to these settings is presented in great detail in the first chapter. After this, the wave front set defined and its connection to singularities of distributions is explained. Among the most important results is the detailed proof of the fact that a distribution which is defined on the target space of a smooth map, and has a suitable wave front set, can be pulled back to a distribution on the domain. The pullback map is shown to be sequentially continuous but not topologically continuous in general. Aided by the pullback map we show how the product of two distributions can be defined when their wave front sets are compatible in a certain way. An application to the theory of PDEs is also given. Lastly, oscillatory integrals are defined and a description of their wave front sets is given.