In cosmology, there's a parameter called "curvature". If the curvature were slightly less than 1, the topology would be "closed", meaning it would be finite, and traveling in a straight line ends back where you started. This is like the surface of the Earth, but in 3 dimensions instead of 2. Triangle angles add up to greater than 180 degrees. If the curvature were equal to 1, the universe would be "flat", meaning it would be infinite and triangles would add up to 180. If the curvature were greater than 1, it would be "hyperbolic", meaning it would be infinite and triangles would add up to less than 180.
If I recall correctly, currently the best measurement of the curvature is 1 +/- .02, so it's probably flat. What's interesting is if the shape was flat and the universe were truly infinite, there would be an infinite number of perfect copies of Earth, the closest of which would be within 10^10^23 meters, as noted by Max Tegmark (source: http://space.mit.edu/home/tegmark/crazy.html)
If I recall correctly, currently the best measurement of the curvature is 1 +/- .02, so it's probably flat. What's interesting is if the shape was flat and the universe were truly infinite, there would be an infinite number of perfect copies of Earth, the closest of which would be within 10^10^23 meters, as noted by Max Tegmark (source: http://space.mit.edu/home/tegmark/crazy.html)