In this paper, we introduce a new class of planar arrays that we call the Bessel planar arrays. A formula for the current distribution in the elements of these arrays is presented, which is related to Bessel functions. For the Bessel planar arrays, the maximal sidelobe level is controllable, the directivity is very high, and the half-power beam width is slightly larger compared to the optimal Chebyshev planar arrays. Methods to set the maximal sidelobe level and compute the directivity and the half-power beam width are described, and numerical examples are given to illustrate the features of the proposed arrays.