It is well-known that any locally Lebesgue integrable function generates a unique distribution, a so-called regular distribution. It is also well-known that many non-integrable functions can be regularized to give distributions, but in general not in a unique fashion. What is not so well-known is that to many distributions one can associate an ordinary function, the function that assigns the distributional point value of the distribution at each point where the value exists, and that in many cases this ordinary function determines the distribution in a unique fashion. In this talk we consider several classes of distributions that are given in terms of the ordinary function of their point values. In particular, we consider distributions that have a point value everywhere, those that have lateral limits at each point, and then introduce the class of distributionally integrable functions. We study several properties of distributions of these classes and apply these ideas to study the boundary behavior of solutions of partial differential equations.