  In each manifold $M$ modeled on a finite or infinite dimensional cube
$[0,1]^n$ 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 $h(A)\subset X$. We also prove that any
two universal meager $F_\sigma$-sets in $M$ are ambiently homeomorphic.
