Andrew Wiles proved Fermat’s last theorem in 1995, using the connection between elliptic curves and modular forms. Actually, Wiles proved a special case of the modularity theorem. The goal of this thesis was to demonstrate application of the modularity theorem and theories of elliptic curves and modular forms to Diophantine equations. In the first chapter we give an overview of basic terms related to elliptic curves. We especially deal with reduction of elliptic curves modulo \(p\): we describe the cases and subcases of good and bad reduction and define the conductor of an elliptic curve. We show an example of finding the minimal equation and conductor. Furthermore, we define isogenies of elliptic curves and give a few important examples of isogenies. The chapter ends with the statement of Mazur’s theorem on \(p\)-isogenies. In the second chapter we define the modular group and its congruence subgroups and then also define modular forms with respect to congruence subgroups. Subsequently, we define newforms: a special class of modular forms, which are especially important for the modularity theorem, and we give an overview of basic properties of newforms. In the end of this chapter we state Ribet’s theorem and the modularity theorem, which give the basis for application of this theory on solving Diophantine equations. In the final chapter we demonstrate the application of the modularity and Ribet’s theorem to solving Diophantine equations. The first example is Fermat’s eqautions \(a^p+b^p+c^p=0\), i.e. we show that Fermat’s last theorem follows from modularity theorem and Ribet’s theorem. After that we look at the equation \(x^p+L^ry^p+z^p=0\) and in that example we show a few methods that can be applied to such equations. One approach is the so called method of Kraus, and for this method we give one more example.