In this paper we present a number of algorithms for two-dimensional (2-D) adaptive beamforming with planar array antennas. These are: (1) the 2-D LMS, (2) the 2-D Howells-Applebaum algorithm, and (3) the 2-D QRD-LS algorithm. In the latter case we also present an architecture for a systolic array processing. The derivation of the first two algorithms is based on the classical methods of gradient descent and control-loop, respectively, which were developed initially for 1-D arrays. The relationship between the 2-D and 1-D adaptive arrays will be discussed. The concept of 2-D eigenvector beams is also presented. The paper shows that an eigenbeam for a 2-D array can be formed by using two independent 1-D eigenbeams which are operating on row / and column m of the array. As a consequence, this leads to the introduction of a 3-D systolic array for performing the QRD-LS minimization. When the data flow is in time-skewed format, adaptations along rows and columns of the 2-D array can be carried out simultaneously.