A typical problem in enumerative combinatorics is to count the size of a set depending upon a positive integer q. Often the result is a polynomial in q (e.g., the chromatic polynomial of a graph), and sometimes a quasi-polynomial. Generally speaking, a quasi-polynomial is a generalization of polynomials, of which the coefficients may not come from a ring but instead are periodic functions with integral periods. Another way to think of a quasi-polynomial is that it is made of a bunch of polynomials, called the constituents. This lecture series aims at introducing the concept of characteristic quasi-polynomials of integral hyperplane arrangements due to Kamiya-Takemura-Terao (2008), and exploring the related areas. In the simplest setting, when a finite set A of integral vectors in Z^n is given, we may naturally associate to it an integral hyperplane arrangement A(R) in the real vector space R^n. We may also consider its q-reduction for any positive integer q and get an arrangement A(Z/qZ) of subgroups in the finite cyclic group (Z/qZ)^n. The central result in the theory states that the cardinality of the complement of A(Z/qZ) is actually a quasi-polynomial in q. This is called the characteristic quasi-polynomial of A as a result of the fact that its first constituent agrees with the characteristic polynomial of A(R). The lecture series consists of three main parts: 1. The constituents and arrangements over abelian groups 2. Connection to Ehrhart theory and root systems 3. Free hyperplane arrangements