  This paper proves the following results: Besides parallelograms and centrally
symmetric hexagons, there is no other convex domain which can form a two-,
three- or four-fold lattice tiling in the Euclidean plane. If a centrally
symmetric octagon can form a lattice multiple tiling, then the multiplicity is
at least seven. However, there are decagons which can form five-fold $($or
six-fold$)$ lattice tilings. Consequently, whenever $n\ge 3$, there are
non-parallelohedral polytopes which can form five-fold lattice tilings in the
$n$-dimensional Euclidean space.
