  G\"{o}del universe is a homogeneous rotating dust with negative $\Lambda$
which is a direct product of three dimensional pure rotation metric with a
line. We would generalize it to higher dimensions for Einstein and pure
Lovelock gravity with only one $N$th order term. For higher dimensional
generalization, we have to include more rotations in the metric, and hence we
shall begin with the corresponding pure rotation odd $(d=2n+1)$-dimensional
metric involving $n$ rotations, which eventually can be extended by a direct
product with a line or a space of constant curvature for yielding higher
dimensional G\"{o}del universe. The considerations of $n$ rotations and also of
constant curvature spaces is a new line of generalization and is being
considered for the first time.
