Diophantine \(D(n)-m\)-tuple is a set of \(m\) positive integers such that a product of each two distinct elements of a set increased by \(n\) is a perfect square. The case where \(n = 1\) is the most studied one, but the case where \(n = 4\) is also interesting, especially because of similar properties between the two cases. In the first three chapters, we will present some new results concerning \(D(4)-m\)-tuples. In the first and second chapter we proved which upper bounds hold for some arithmetical sums that can be used to estimate a number of possible \(D(4)\)-pairs in a given interval and we studied a problem of extending a \(D(4)\)-triple to a \(D(4)\)-quadruple and properties of elements of binary recurrence sequences associated with that extension. In the third chapter we present a proof of nonexistence of \(D(4)\)-quintuples, which is a central part of this thesis by its significance and size of the content. In the last chapter, one problem concerning a polynomial variant of the problem of Diophantus is presented. Instead of considering sets of positive integers, we observe sets containing polynomials with integer coefficients for which it is usually possible to prove some similar results as in the integer case. We have considered an extension of polynomial \(D(-1)\)-triple \(\{a, b, c\}\) with an element d such that polynomials \(ad + 1, bd + 1\) and \(cd + 1\) are perfect squares of polynomials with integer coefficients and we proved that such \(d\) must be unique under some conditions.