How fast can a spaceship fly in space today, and why can't they fly faster?

The fastest spacecraft ever built flew at about 70 km/second. The limit is set by how much fuel you can carry. Remember, the fuel burning now has to accelerate not only the payload but also all the as yet unburnt fuel.We'll only get to speeds close to light when we can build much more efficient ion rockets, which have a much higher exhaust velocity than chemical rockets, or nuclear powered rockets, which carry far more stored energy than chemical fuel. The Daedalus project, suggested back in the 1960s, would accelerate a spacecraft to about 1/5 the speed of light by detonating a series of tiny H bombs behind it.