We present the first fourth-order, central scheme for two-dimensional hyperbolic systems of conservation laws. Our new method is based on a Central Weighted Non-Oscillatory (CWENO) approach. The heart of our method is the reconstruction step, in which a genuinely two-dimensional interpolant is reconstructed from cell-averages by taking a convex combination of building blocks in the form of bi-quadratic polynomials. Similarly to other central schemes, our new method enjoys the simplicity of the black-box approach. All that is required in order to solve a problem is to supply the flux function and an estimate on the speed of propagation. The high-resolution properties of the scheme as well as its resistance to mesh orientation, and the effectiveness of the component-wise approach, are demonstrated in a variety of numerical examples.