We present accurate estimates for the prefactors of the second and third moments of the height and free-energy fluctuations, as well as the leading correction to the growth rate and free energy per unit length, obtained from extensive simulations of a wide range of one-dimensional models of growing interfaces and directed polymers in a random environment. When scaled by the appropriate model-dependent parameters the amplitudes reduce to universal numbers which characterize the strong-coupling fixed point of the equation of Kardar, Parisi, and Zhang [Phys. Rev. Lett. 56, 889 (1986)]. To check the expected scaling we use models for which the model parameters can be computed analytically. For other systems, such as ballistic deposition, the restricted solid-on-solid model, and the finite-temperature directed polymer, the parameters are determined numerically from steady-state properties. Apart from the standard transient simulation which starts from a flat interface, we also report results for time-dependent correlations in the steady state, which give rise to different universal amplitudes. We compare our results with recent predictions arising from replica calculations and dynamic renormalization-group treatments, finding agreement in the latter but severe discrepancies in the former case. We speculate that the failure of replica theory may be indicative of replica symmetry breaking. © 1992 The American Physical Society.