At the beginning of this thesis, we prove the Weierstrass Approximation Theorem which says that the space of algebraic polynomials is dense in \(C[0,1]\). The second chapter is devoted to the proof of the main result in this thesis, which is the Müntz-Szász Theorem that characterizes sets of powers \(\Pi(\Lambda)=\mathrm{span}\{x^{\lambda_k}:k \in \mathbb{N}_0,\; 0=\lambda_0<\lambda_1<\cdots\}\) that span a dense subset in the space of continuous functions on the segment \([0,1]\). In the third chapter we consider a more general case of the theorem called Full Müntz Theorem. Here we are also interested in the density of the set \(\Pi(\Lambda)\) in some space of functions, but the difference from the initial theorem is that here we aren’t given an increasing sequence of real numbers, but instead we are working with arbitrary sequences of real numbers. Most of the chapter is taken up by the proof of the Full Müntz Theorem for spaces of functions \(C[0,1]\) and \(L^p[0,1]\), for \(L^p[0,1]\). After that, the Full Müntz Theorem for \(L^p[0,1]\), where \(0<a<b\), is examined, and at the very end a few results of the Full Müntz Theorem for \(C(A)\) are listed, where \(A\subset [0,\infty)\) is a set of positive Lebesgue measure.