We consider the problem of direction of arrival (DOA) estimation using a newly proposed structure of non-uniform linear arrays, referred to as co-prime arrays, in this paper. By exploiting the second order statistical information of the received signals, co-prime arrays exhibit O(MN) degrees of freedom with only M + N sensors. A sparsity based recovery method is proposed to fully utilize these degrees of freedom. Unlike traditional sparse recovery methods, the proposed method is based on the developing theory of super resolution, which considers a continuous range of possible sources instead of discretizing this range into a discrete grid. With this approach, off-grid effects inherited in traditional sparse recovery can be neglected, thus improving the accuracy of DOA estimation. In this paper we show that in the noiseless case one can theoretically detect up to M N sources with only 2M + N sensors. The noise 2 statistics of co-prime arrays are also analyzed to demonstrate the robustness of the proposed optimization scheme. A source number detection method is presented based on the spectrum reconstructed from the sparse method. By extensive numerical examples, we show the superiority of the proposed method in terms of DOA estimation accuracy, degrees of freedom, and resolution ability compared with previous methods, such as MUSIC with spatial smoothing and the discrete sparse recovery method.