In this talk we will discuss bounded weight modules, i.e., modules that decompose as direct sums of weight spaces and whose sets of weight multiplicities are uniformly bounded. Our main focus will be on the direct limits of classical Lie (super)algebras. In particular, we will present the classification of the simple bounded weight modules over $\mathfrak{sl} (\infty)$, $\mathfrak{o} (\infty)$, $\mathfrak{sp} (\infty)$, as well as over their super-analogs. A key role in the study plays the theory of weight modules over Weyl and Clifford superalgebras of infinitely many variables. The talk is based on joint works with I. Penkov and V. Serganova.