The nonlinear stress and microstructural response of a colloidal hard sphere suspension undergoing medium and large amplitude oscillatory simple shear have been studied using Accelerated Stokesian dynamics. The goal is to understand how nonlinearity arises and to link the structural effects to the resulting suspension stress. The imposed shear is given by the time-dependent rate (gamma)over dot(t) = (gamma)over dot(i alpha t) . Most results are shown for a hard-sphere suspension at a particle volume fraction phi = 0.4. These are freely flowing conditions far from either glassy or jammed conditions, but the concept of the particle cage from glass dynamics is used. The cage is defined here in a statistical manner as the surface of elevated nearest neighbor probability, a sphere at contact for equilibrium. The cage concept is used in interpreting the microstructural deformation: For sufficiently small strain amplitude gamma(0), the cage deforms negligibly due to flow and the suspension remains in the linear response regime, but this is found to require gamma(0) < 0: 01 at phi = 0.4, as shown by a spectral decomposition of the microstructure in time, which discriminates rigorously between linear and nonlinear deformation. At larger gamma(0), termed medium amplitude and large amplitude in other studies, the material response is nonlinear. To preface the large amplitude oscillatory shear analysis, we use linear viscoelasticity theory to compare stress fluctuations at equilibrium to results obtained at finite Peclet number Pe and small gamma(0), as well as available experimental data and theoretical predictions; Pe = 6 pi eta(gamma)over dot(0)a(3)/kT is the ratio of hydrodynamic to Brownian forces, where g is the viscosity of the suspending liquid, (gamma)over dot(0) is the shear rate amplitude, a is the particle radius, k is the Boltzmann constant, and T is the absolute temperature. The shear stress sigma(xy) and the normal stress differences N-1 and N-2 are analyzed under oscillatory shear at amplitudes 0.01 <= gamma(0) <= 3.6 for a range of Pe. (The frequency alpha is related to Pe through (gamma)over dot = alpha gamma(0) and the nondimensional frequency is given by the Deborah number De = Pe/gamma(0) = 6 pi eta alpha a(3)/kT.) Pipkin diagrams are shown for sigma(xy), N-1 and N-2. When hydrodynamic forces dominate the flow of the suspension, the complex viscosity vertical bar eta*vertical bar has a nonmonotonic dependence on gamma 0, and Fourier-transform rheology shows the nonlinearity of the stress response to be maximized at an intermediate strain amplitude that depends on Pe. The elastic and viscous behavior of the suspension, as determined by a Chebyshev polynomial decomposition, is distinctly different for small and large Pe. The influence of the microstructure on the normal stress differences is discussed, noting that N-1 is significant only when angular distortion of the microstructure is present, whereas N-2 is formed with an accumulation of pair correlation at contact even at low oscillation amplitude. (C) 2017 The Society of Rheology.