  Orthogonal polynomials are of fundamental importance in many fields of
mathematics and science, therefore the study of a particular family is always
relevant. In this manuscript, we present a survey of some general results of
the Hermite polynomials and show a few of their applications in the connection
problem of polynomials, probability theory and the combinatorics of a simple
graph. Most of the content presented here is well known, except for a few
sections where we add our own work to the subject, nevertheless, the text is
meant to be a self-contained personal exposition.
