In this paper, the direction of arrival (DOA) and orientation of array (OOA) estimation problems in the recently proposed orientational beamforming (OBF) system are investigated. We show that existing DOA estimation methods can be directly applied to estimate the DOA information in the OBF system. Therefore, this paper mainly focuses on the OOA estimation problem. Three OOA estimation methods are first proposed based on the assumption of accurate DOA estimation. In each method, the OOAs of different transmitting arrays are estimated individually by searching in the one-dimensional parameter space. If the power spectral density (PSD) of the source signal emitted from a transmitting array is known, the OOA of the transmitting array can be estimated by the proposed PSD-based least squares method. Otherwise, the electronical rotation-based OBF (ER-OBF) and single-source-array OBF (SSA-OBF) methods are proposed. In both methods, the OOA of a transmitting array is estimated by detecting the peak in its corresponding orientational beampattern obtained by electronically rotating the receiving array. The electronical rotation of the receiving array is achieved by appropriately delaying received signals according to the DOA of the source signal. Based on the proposed ER-OBF method, the SSA-OBF method further proposes to use a transform matrix to suppress the mutual interference between signals from different transmitting arrays. Simulation results show that the probability of successful estimation of the proposed methods increases as the signal bandwidth and array aperture size increase. Moreover, observing that the DOA estimation error impacts the performance of the proposed OOA estimation methods, a modified random search method is developed to achieve OOA estimation in the presence of DOA estimation error.