A harmonic balance technique for modeling unsteady nonlinear flows in turbomachinery is presented. The analysis exploits the fact that many unsteady flows of interest in turbomachinery are periodic in time. Thus, the unsteady flow conservation variables may be represented by a Fourier series in time with spatially varying coefficients. This assumption leads to a harmonic balance form of the Euler or Navier-Stokes equations, which, in turn, can be solved efficiently as a steady problem using conventional computational fluid dynamic (CFD) methods, including pseudotime time marching with local time stepping and multigrid acceleration. Thus, the method is computationally efficient, at least one to two orders of magnitude faster than conventional nonlinear time-domain CFD simulations. Computational results for unsteady, transonic, viscous flow in the front stage rotor of a high-pressure compressor demonstrate that even strongly nonlinear flows can be modeled to engineering accuracy with a small number of terms retained in the Fourier series representation of the flow. Furthermore, in some cases, fluid nonlinearities are found to be important for surprisingly small blade vibrations.