Wave profiles were measured in an Fe–Cr–Ni alloy (stainless steel 304) shock compressed to Hugoniot stresses between 7 and 80 GPa. A single-stage propellant gun was used to generate shock states and time histories were recorded by velocity interferometry. The particle velocity measurements are generally consistent with impedance match calculations to ±2%. Unloading wave velocities were obtained from analysis of the release wave profiles. Using Eulerian finite strain theory and under the assumption of fully elastic initial release, the first and second pressure derivatives of the longitudinal modulus are given by: 7.9(0.5) and –0.16(0.06) GPa–1, where the numbers in parentheses are one standard deviation uncertainties. The first and second pressure derivatives of the adiabatic bulk modulus are: 6.4(1.0) and –0.17(0.08) GPa–1. The unloading wave velocities are generally consistent with extrapolated trends from low-pressure ultrasonic data as well as with higher stress shock measurements on an alloy of similar composition. From 1 bar to 80 GPa, Poisson's ratio, nu, increases with Hugoniot stress, sigma (in GPa), according to the relation: nu =0.29 + 0.0008 sigma. The Hugoniot elastic limit of 304 steel was found to be 0.35(0.12) GPa, and the initial yield stress is 0.21(0.07) GPa. The elastic precursor velocity was 5.8(0.1) km/s. Numerical simulations of the wave profiles using a constitutive model that incorporates a Bauschinger effect and stress relaxation reproduced the main features observed in the profiles. Release adiabats were also calculated from the measured wave profiles. The shear stress at unloading was determined to vary with stress according to the relation: tau0+tauc=0.149+0.018 sigma, where sigma is given in GPa.