Millisecond X‐ray time variability studies of accreting low‐magnetic‐field neutron stars in X‐ray binaries probe the motion of matter in regions of strong gravity. In these regions, general relativity (GR) is no longer a small correction to the classical laws of motion, but instead dominates the dynamics: we are studying motion in strongly curved spacetime. Such millisecond X‐ray variability studies can therefore provide unique tests of GR in the strong‐field regime. The same studies also constrain neutron‐star parameters such as stellar mass and radius, and thereby the equation of state (EOS) of ultradense matter. By comparing neutron star and black hole phenomenology the models can be constrained in unique ways. I discuss the prospects for mapping out space‐time near accreting stellar‐mass compact objects, and measuring the EOS of dense matter, through millisecond timing, particularly with an eye towards future missions. The key to further progress is timing sensitivity, and the overwhelming consideration for timing sensitivity is collecting area: contrary to most applications, the signal‐to‐noise ratio for the aperiodic timing phenomena produced by accretion flows increasesproportionallywith count rate rather than as the square root of it. A ten times larger instrument turns 1&sgr; effects into 10&sgr; effects (or does as well in 1&percent; of the time). With the Rossi X‐ray Timing Explorer (RXTE), using 0.6 m2collecting area, we have found several timing diagnostics from the accretion flow in the strong‐gravity region around neutron stars and black holes. Next‐generation timing instruments, larger in area by an order of magnitude and with enhanced spectral capabilities, are expected to turn thesediagnosticsof GR into truetestsof GR. They are also expected to put strong constraints on neutron‐star structure, and thereby on the EOS of supranuclear density matter. © 2004 American Institute of Physics