The structure of a shock wave interacting with isotropic turbulence is investigated. General principles of similarity scaling show that consistency with known physical limiting behavior requires incomplete similarity solutions where the governing non-dimensional parameters, namely, the Reynolds, convective, and turbulent Mach numbers (Rλ, M, and Mt, respectively), can be combined to reduce the number of similarity parameters that describes the phenomenon. An important parameter is found to be K = Mt/Rλ1/2(M − 1) which is proportional to the ratio of laminar shock thickness to the Kolmogorov length scale. The shock thickness under turbulent conditions, on the other hand, is essentially a random variable. Under a quasi-equilibrium assumption, shown to be valid when K2 ≪ 1, analytical results are obtained for the first and second moments of the turbulent shock thickness, velocity gradient, and dilatation at the shock. It is shown that these quantities exhibit universal behavior in the parameter K with corrections in Mt/(M − 1), for velocity fields with arbitrary statistics. Excellent agreement is observed with available data from direct numerical simulations. Two-point statistics of velocity gradients at the shock show that the distribution of dilatation over the shock surface is determined by transverse structure functions of the incoming turbulence. The regimes of the interaction are also investigated. It is found that the appropriate parameter to delimit the different regimes is Mt/(M − 1). Flow retardation ahead of the shock is suggested as a mechanism for so-called broken shocks.