The microwave theory of phased-array antennas is reviewed, emphasizing large planar arrays suitable for phased-array radar. The review looks in detail at the three major parts of the antenna, namely, the element array, the phase shifters, and the feed system. The element array is studied by first modeling it as being infinite in extent and the concept of the per element absorption cross section and gain versus angle is developed. The technique of modal analysis using plane waves is explained and used to calculate the aforementioned quantities for waveguide and dipole elements. The review proceeds first through an analysis which applies to small elements, using a single element mode, and then to a more accurate solution containing the first higher mode in addition. The author has added new research to the review which explains the blindness effect for waveguide arrays. A necessary and sufficient condition for array blindness has been derived which removes a major uncertainty about the blindness effect. A result is that a blindness angle will always occur in E and H plane scanning of rectangular waveguide elements for any waveguide size, provided the array lattice is such as to permit a grating-lobe singularity by the simple one-element-mode theory. Elements large than some critical size are not required to produce blindness but the large size has the effect of producing greater shift of the blindness angle towards broadside from the grating lobe angle. The singularity in element admittance right at the grating lobe angle, which is a well-known milestone in the analysis of elements which have been modeled to support only one mode, is in reality a fiction. When higher modes are added to the solution, as is necessary for any physical element, the grating-lobe singularity is found to disappear. Thus for any real physical element, there is no grating lobe blindness, but the blindness angle is shifted inside of this angle by an amount which depends upon the element size. A review of element design calculations using many higher element modes is presented, and element configurations are shown which produce a good impedance match over a wide scan angle as well as a wide-frequency band. The application of the aforementioned results obtained from the infinite-array model to a practical, finite array are discussed. The theory of the most useful phase-shifter types for phased arrays is discussed, including those using both ferrite and semiconductor diodes. The ferrite toroid in a waveguide is a very effective design and the results of an optimization analysis for this configuration are presented. Choices of ferrite material parameters for low loss at both small and large signal levels are discussed. A method of reducing the temperature sensitivity of phase shift by means of the driver circuit is also reviewed. Semiconductor diode phase shifters have a greatly increased potential due to an improvement in diode reliability that has produced an expected mean life of 109h. The theory of phase-shifter operation using diodes in a balanced hybrid circuit is discussed and relations are given for bandwidth, loss, and power capability. The high-power limitation of the phase shifters is due to a nonlinear loss effect in the reverse biased state and is caused by the large RF voltages. The loss can be reduced by circuit techniques and by employing the new diodes of improved design, which are discussed. Feed systems for phased arrays are synthesized from a number of basic techniques according to the antenna applications. A few of the important techniques are reviewed here. Feed systems are required to produce an optimum aperture distribution for two types of patterns simultaneously, the sum pattern (Taylor distribution) and the difference pattern (Bayliss distribution). For the constrained feed, which uses transmission line throughout, two alternatives are presented for accomplishing the aforesaid. It is shown that the feed network can prevent reflections from the element array, which are always present to some degree, from adding an error component to the desired aperture distribution by employing 4-port power dividers which are provided with a reflection absorbing termination. An approximate method of wide-band beam steering is discussed in which a small number of time delay devices are used to feed subarrays of elements steered by conventional phase shifters. The subarray feeding technique is also employed in another application with the Butler beam forming matrix to form simultaneous multiple beams in an approximate manner that reduces the size of the beam forming matrix required. The space feed is an alternative to the constrained feed and distributes energy to the elements by free-space propagation. This method possesses great flexibility and may approach the capability of the constrained feed in secondary aperture performance by using a large number of feed elements. A design method of accounting for the near-field diffraction produced by the large feed is given.