Let $G=\GL(2, \mathbb{F}_q)$ denote the group of invertible $2\times 2$ matrices over the finite field $\mathbb{F}_q$, where $q$ is the power of a prime number. This thesis investigates the complex irreducible representations of $G$. Chapter 2 gives an overview of the general theory of finite-dimensional representations of a finite group over the complex numbers. In Chapter 3, we discuss the different types of conjugacy classes in $G$. It will turn out that there are four types, which altogether give $q^2-1$ conjugacy classes. By general theory, the number of irreducible representations distinguished up to isomorphism is also equal to $q^2-1$. These representations are explicitly constructed in Chapter 4. Some of them are derived from the permutation representation of $G$ on the projective line $\mathbb{P}^1(\FF_q)$, while the rest are induced from smaller subgroups of $G$. In the concluding chapter, we compute the irreducible decompositions of the tensor products of some of these irreducible representations by using the character table for irreducible $G$-representations.