  In this paper we show that one can have asymptotically de Sitter (dS),
anti-de Sitter (AdS) and flat solutions in Gauss-Bonnet gravity without any
need to a cosmological constant term in field equations. First, we introduce
static solutions whose 3-surfaces at fixed $r$ and $t$ have constant positive
($k=1$), negative ($k=-1$), or zero ($k=0$) curvature. We show that for
$k=\pm1$, one can have asymptotically dS, AdS and flat spacetimes, while for
the case of $k=0$, one has only asymptotically AdS solutions. Some of these
solutions present naked singularities, while some others are black hole or
topological black hole solutions. We also find that the geometrical mass of
these 5-dimensional spacetimes is $m+2\alpha | k| $, which is different from
the geometrical mass, $m $, of the solutions of Einstein gravity. This feature
occurs only for the 5-dimensional solutions, and is not repeated for the
solutions of Gauss-Bonnet gravity in higher dimensions. We also add angular
momentum to the static solutions with $k=0$, and introduce the asymptotically
AdS charged rotating solutions of Gauss-Bonnet gravity. Finally, we introduce a
class of solutions which yields an asymptotically AdS spacetime with a
longitudinal magnetic field which presents a naked singularity, and generalize
it to the case of magnetic rotating solutions with two rotation parameters.
