For Near-Earth Orbits (and sometimes farther), telemetry data known as Two Line Elements(TLEs) are commonly used to propagate orbits. Propagating an orbit is a term that means predicting future locations from a known state. TLEs look like:
HST 1 20580U 90037B 22010.54856767 .00000893 00000-0 42745-4 0 9993 2 20580 28.4696 134.5382 0002786 129.9204 289.6315 15.09931360542325
Hubble Space Telescope (HST)
There are a number of places to look up recent TLE sets on almost anything orbiting Earth.
A searchable SATellite CATalog:
TLE’s are valid for propagating orbits with decent precision for a few days after their original Epoch.