Beamforming is traditionally associated with coherent summation of signals from antenna elements of the same polarization, here referred to as single polarization beamforming (SPBF). In this paper we focus on a new method, called dual polarization beamforming (DPBF), to design beam patterns. Instead of using only a single element-polarization, as for SPBF, a desired beam pattern is designed as the sum of powers for two orthogonal element-polarizations. So, with DPBF the focus is on total radiated power beam patterns. The DPBF technique provides additional degrees of freedom to form a desired beam pattern such that amplitude variations in the beamforming vector can often be significantly reduced, potentially to uniform amplitude. Phase-only beamforming, possibly using only minor amplitude variations, is a very interesting property, especially for active antennas, since it offers the potential of full power amplifier utilization. In this paper we apply DPBF to uniform linear arrays (ULAs) as well as uniform rectangular arrays (URAs). In communication systems, it is often desired to have a second beam with identical beam pattern but orthogonal polarization in all directions. We show how such beams are designed with DPBF, both for ULAs and URAs.