The purpose of the present paper is to establish moment estimates of Rosenthal type for a rather general class of random variables satisfying certain bounds on the cumulants. We consider sequences of random variables which satisfy a central limit theorem and estimate the speed of convergence of the corresponding moments to the moments of a standard normally distributed variable. The examples of random objects we discuss include those where a dependency graphs or a weighted dependency graph encodes the dependency structure. We give applications to subgraph-counting statistics in Erdős-Rényi random graphs of type $G(n,p)$ and $G(n,m)$, crossings in uniform random pair partitions and spins in the $d$-dimensional Ising model. Moreover, we prove moment estimates for certain statistics appearing in random matrix theory, namely characteristic polynomials of random unitary matrices as well as the determinants of certain random matrix ensembles. We add estimates for the $p(n)$-dimensional volume of the simplex with $p(n)+1$ points in ${\mathbb R}^n$ distributed according to special distributions, since it is strongly connected to Gram matrix ensembles.