Ambient energy harvesting has been in recent years the recurring object of a number of research efforts aimed at providing an autonomous solution to the powering of small-scale electronic mobile devices. Among the different solutions, vibration energy harvesting has played a major role due to the almost universal presence of mechanical vibrations: from ground shaking to human movements, from ambient sound to thermal noise. Standard approaches are mainly based on resonant linear oscillators that are acted on by ambient vibrations. Here we propose a new method based on the exploitation of the dynamical features of stochastic nonlinear oscillators. Such a method is shown to outperform standard linear oscillators and to overcome some of the most severe limitations of present approaches, like narrow bandwidth, need for continuous frequency tuning and low efficiency. We demonstrate the superior performances of this method by applying it to piezoelectric energy harvesting from ambient vibration. Experimental results from a toy-model oscillator are described in terms of nonlinear stochastic dynamics. We prove that the method proposed here is quite general in principle and could be applied to a wide class of nonlinear oscillators and different energy conversion principles. There are also potentials for realizing micro/nano-scale power generators.