We will talk about classical problems such as representing a given integer as sum of two squares, or cubes and generalization of such natural questions. We will survey some classical facts in Number Theory on approximation of algebraic numbers by rationals. We will talk about how these wonderful results in the field of Diophantine approximation lead to interesting theorems about Diophantine equations. For example, we will see that if F(x,y) is an irreducible binary form with integer coefficients of degree at least 3, then the equation F(x,y) = 1 can have only finitely many solutions in integers x,y. We will conclude by stating more recent results about counting the number of integer solutions of such equations.