Suprathermal particles are ubiquitously present in the solar wind, with energies typically from ~1 keV nucleon-1 to ~a few MeV nucleon-1. Remarkably, the suprathermal particles exhibit a common spectral shape in many different circumstances; the distribution function is a power law in particle speed, with spectral index of -5. The observations cannot be explained by traditional stochastic acceleration, which yields spectra that depend on, e.g., the momentum diffusion coefficient, and thus the spectra are expected to be different in different circumstances. A theory is presented in which the particles are accelerated in thermally isolated compressional turbulence. Thermal isolation is valid in spatially homogeneous conditions in the solar wind, and when properly applied, yields suprathermal tails that always have the required spectral shape. The theory describes the time evolution of the spectrum to its equilibrium form and predicts the high-speed cutoff on the acceleration. The high-speed cutoff is shown to be in good agreement with observations of quiet-time spectra at Earth and is consistent with observations throughout the outer heliosphere.