There are three main possibilities for the shape of the universe but because the universe is expanding time is running out to detemine its shape:
Sphere
A spherical universe has positive curvature: It is finite in size, but without boundaries, like a balloon.
In a so-called closed universe, you could, in principle, fly a spaceship in one direction and eventually get back to where you started from.
A closed universe is also closed in time: It eventually stops expanding, then contracts in a "Big Crunch."
In such a universe, parallel lines eventually converge (e.g. longitudinal lines are parallel at the equator, but converge at the poles) and large triangles have more than 180 degrees.
Flat
You can imagine this kind of universe by cutting out a piece of balloon material and stretching it with your hands. The surface of the material is flat, not curved, but you can expand and contract it by tugging on either end.
A flat universe is infinite in size, and has no boundaries.
In such a universe, parallel lines are always parallel and triangles always have 180 degrees.
A flat universe expands forever, but the expansion rate approaches zero.
Saddle
Such a universe has negative curvature: It is infinite and unbounded.
In a so-called open universe, parallel lines eventually diverge, and triangles have less than 180 degrees.
An open universe expands forever, with the expansion rate never approaching zero.
http://www.theglobeandmail.com/servlet/ArticleNews/TPStory/LAC/20040710/UNIVERSE10/TPScience/