Adaptive arrays are a radical departure from conventional thinking in antenna design, offering substantial improvements in performance over fixed pattern antennas in environments that include severe interference and jamming. They achieve this because they are designed to steer nulls automatically at noise sources of unknown or variable direction and generally to modify their beampatterns to optimise performance. Adaptive array processing is applicable in most systems that exploit wave propagation; typical uses being radar, active and passive sonar, radio communication links, and radio monitoring. Although sensors and hardware for different applications vary, the same optimality criteria are used throughout and similar algorithms may be employed. This book develops the concepts underlying the design of adaptive arrays from first principles and is directed at research workers and designers whose mathematical background requires refurbishment in the special techniques that have accumulated around the field, often to the obscuration of the simple basic ideas. The topics treated include: single multiple null steering; derivation of the weighting coefficients in an array that maximises signal to noise ratio; online algorithms for achieving these coefficients using gradient methods based on correlators and coefficient peterbation; direct estimation of optimum coefficients by covariance matrix inversion and recursive techniques; prevention of null steering at the desired source and control over the main lobe shape; minimisation of the number of variable coefficients in suboptimal implementations.