In this thesis we have studied the notion of entropy solutions of first-order quasilinear partial differential equations (PDEs), and have shown their existence and uniqueness. Firstorder quasilinear PDEs often occur in mathematical models because the conservation laws belong to this group of equations. As in general such equations are not smooth enough, weak solutions, ie. the corresponding subclass of entropy solutions are observed. The first chapter provides an overview of the theory for solving the Hamilton-Jacobi equation and the corresponding initial problem, and lists some technical statements that we use later. The second chapter deals with scalar conservation laws as the simplest case of a quasilinear equation. An entropy condition is introduced here by which the notion of an entropy solution is defined. Then, the existence and uniqueness of such a solution was demonstrated using the Lax-Oleinik formula. Finally, the developed theory was applied to the Riemann problem. The third chapter illustrates the equivalent formulation of the entropy solution. The notion of entropy solution is studied through the concept of entropy and entropy flux. Entropy inequality is introduced as a way of recognizing physical solutions among all weak solutions. In the end, we solved the Riemann problem again using this approach and, as expected, obtained the same entropy solution.