In each manifold $M$ modeled on a finite or infinite dimensional cube ${[0,1]}^n$, $n\le \omega$, we construct a meager $F_{\sigma }$-subset $X\subset M$ which is universal meager in the sense that for each meager subset $A\subset M$ there is a homeomorphism $h:M\to M$ such that $h\left(A\right)\subset X$. We also prove that any two universal meager $F_{\sigma }$-sets in $M$ are ambiently homeomorphic.