In this talk we give an introduction to infinite dimensional moment problems, i.e. for measures supported on infinite dimensional spaces. Although infinite dimensional moment problems have a long history, the theory is still not as well developed as in the finite dimensional case. We will focus on the following problem: when can a linear functional on a unital commutative real algebra A be represented as an integral w.r.t. a Radon measure on the character space X(A) of A equipped with the Borel σ-algebra generated by the weak topology? Our main idea is to construct X(A) as a projective limit of the character spaces of all finitely generated subalgebras of A, so to be able to exploit the classical finite dimensional moment theory in the infinite dimensional case. Thus, we obtain existence results for representing measures defined on the cylinder σ-algebra on X(A), carried by the projective limit construction. If in addition the well-known Prokhorov (ε-K) condition is fulfilled, then we can solve our problem by extending such representing measures from the cylinder to the Borel σ-algebra on X(A). These results allow us to establish infinite dimensional analogues of the classical Riesz-Haviland and Nussbaum theorems as well as a representation theorem for linear functionals nonnegative on a “partially Archimedean” quadratic module of A. Our work applies to the case when A is the algebra of polynomials in infinitely many variables or the symmetric tensor algebra of a real infinite dimensional vector space, providing alternative proofs of some recent results for these instances of the moment problem and offering at the same time a unified setting which enables comparisons. (This is a joint work with Salma Kuhlmann, Tobias Kuna and Patrick Michalski)