Wind turbines generate electricity from turbulent wind. Large fluctuations, and, more importantly, frequent wind gusts cause a highly fluctuating electrical power feed into the grid. Such effects are the hallmark of high-frequency turbulence. Here we show evidence that it is the complex structure of turbulence that dominates the power output for one single wind turbine as well as for an entire wind farm. We illustrate the highly intermittent, peaked nature of wind power fed into the grid. Multifractal scaling is observed, as described initially by Kolmogorov's 1962 theory of turbulence. In parallel, we propose a stochastic model that converts wind speed signals into power output signals with appropriate multifractal statistics. As more and more wind turbines become integrated into our electric grids, a proper understanding of this intermittent power source must be worked out to ensure grid stability in future networks. Thus, our results stress the need for a profound understanding of the physics of turbulence and its impact on wind energy.