The moment problem (MP) is a classical question in analysis that has been studied since the end of the 19th century appearing first in the memoir of the famous Dutch mathematician Stieltjes in 1894. The question was whether a given sequence of numbers can be represented as moments of some distribution on the Borel sigma--algebra on a half-line. After Stieltjes solved the problem, this led to follow-up questions in which the half-line was replaced by the real line (Hamburger, 1920) and by a finite interval (Hausdorff, 1923). Among other famous mathematicians who made important early contributions in the field we mention Carleman, Nevanlinna and Riesz. The fact being especially fascinating about the MP is its interplay with many different areas of mathematics and a broad range of applications, such as operator theory, probability and statistics, inverse problems, numerical analysis and, more recently, real algebraic geometry, polynomial optimization, control theory, partial differential equations and others. A variant of the MP that occurs even more frequently in applications is the one in which only finitely many numbers are given in the sequence, the so-called truncated moment problem (TMP). Apart from the applications, the TMP is also more general than the full version, which is a recent remarkable result of Stochel (2001). A lot of work on univariate TMPs was done in the early second half of the 20th century by Akhiezer, Krein and Nudelman, while in the last three decades renown interest in TMPs started with a series of papers by Curto and Fialkow, leaning on the interplay of TMPs with real algebraic geometry (RAG). RAG studies certificates, called Positivstellensätze, for positivity of polynomials on positivity sets of other polynomials. The beginning of RAG is Hilbert's 17th problem from 1900, which asked whether every positive polynomial is a sum of squares of rational functions, and was answered in the affirmative by Artin in 1926. This led to further generalizations of the problem later in the century. The connection between the MP and RAG is Haviland's theorem from 1935, which states that the MP on a set K has a solution if and only if the corresponding functional, which is defined on the vector space of all polynomials, is positive on positive polynomials. This interplay received new attention with Schmüdgen's solution of the multidimensional MP in 1991, which combines the Positivstellensatz with ideas from functional analysis. However, in order to apply this interplay for the truncated case, one needs Positivstellensätze for strictly positive polynomials with degree bounds. Such certificates are difficult to obtain and this is the reason why concrete solutions are only known in special cases, in particular when K is a quadratic curve. Besides positive semidefiniteness of the moment matrices, there are some additional numerical conditions in concrete solutions that are difficult to obtain. The TMP on plane curves of degree more than two is widely open.The main aims of this proposal are to: (i) solve the TMP on cubic and rational curves completely, (ii) solve the TMP in the two dimensional space and (iii) generalize results from the scalar to the matrix TMPs.