The author gives solvability conditions of moment problems for an arbitrary sequence of linear continuous functionals over Fréchet spaces. As is known many theorems in analysis state the solvability conditions for numerous moment problems. The author notes that the solvability problem is equivalent to the fact that the sequence of linear functionals is linearly independent and that the factor space of some Hausdorff space \(E\) is isomorphic to the space of all numerical sequences with the product topology. The author proves the solvability criterion for moment problems in Fréchet spaces. As applications the moment problem for the sequence of functionals on the space of test functions connected with the order of singularity of the corresponding space of generalized functions is solved. As corollary of his key theorem the author formulates the classical Borel lemma of the existence of a smooth function assuming prescribed values at given points and establishes a series of new solvability theorems for the moment problem in Fréchet spaces.