The thesis is structured into 5 chapters as follows: <strong>Chapter 1</strong> is an introduction to the tools and methods we use most frequently. <strong>Chapter 2</strong> Pairs of k-free Numbers, consecutive square-full Numbers. In this chapter, we refine the approximate determinant method by Heath-Brown. We present applications to asymptotic formulas for consecutive k-free integers, and more generally for k-free integers represented by r-tuples of linear forms. We also show how the method can be used to derive an upper bound for the number of consecutive square-full integers. Finally, we apply the method to make a statement about the size of the fundamental solution of Pell equations. <strong>Chapter 3</strong> Power-Free Values of Polynomials. A conjecture by Erdös states that for any irreducible polynomial f of degree d&ge;3 with no fixed (d-1)-th power prime divisor, there are infinfinitely many primes p such that f(p) is (d-1)-free. We prove this conjecture and derive the corresponding asymptotic formulas. <strong>Chapter 4</strong> Integer Points on Bilinear and Trilinear Equations. In the fourth chapter, we derive upper bounds for the number of integer solutions on bilinear or trilinear forms. <strong>Chapter 5</strong> In the fifth chapter, we present a method to count the monomials that occur in the projective determinant method when the method is applied to cubic varieties.