Depends on how you count them. The answer above (1100) is probably in the right neighborhood for operational satellites. The number of dead satellites, debris objects and rocket bodies is far higher. If you have a Space-Track.org account, take a look under the "Boxscore" for the current count, broken down by country/organization of origin.
Rocket Bodies: 2005
This is the total, as released by the US Air Force, and doesn't include a significant population which aren't publicly released for various reasons. It also doesn't include large swaths of the debris population which is either to small to reliably track, or hasn't been catalogued for various reasons. Depending on where you draw the line for the lower size bound, some estimates are as high as 100,000 objects.