It is widely anticipated that the first direct detections of gravitational waves will be made by advanced gravitational-wave detectors, such as the two Laser Interferometer Gravitational-wave Observatories (LIGO) and the Virgo interferometer. Arguably the most important source for ground-based interferometers are coalescing binary neutron stars. Following the detection of such a system, a more detailed followup analysis will seek to measure certain properties of the component neutron stars, such as their masses and/or spin configurations. In particular, it has been shown that the gravitational waves emitted by binary neutron stars carry information about the neutron-star equation of state. In this talk, I will discuss how gravitational-wave detectors can be used to measure equation-of-state effects in binary neutron star systems. I will then present a new method for directly measuring the neutron-star equation of state by estimating the parameters of a 4-piece polytropic model that matches theoretical equation-of-state candidates to a few percent. I will finish by discussing the prospects of measuring the neutron-star equation of state with the Advanced LIGO-Virgo network. If all goes according to plan, the Advanced LIGO detectors will begin their first official observing run by the time I give this talk!