This review paper discusses the basic properties of two adaptive signal processing procedures for dealing with weak scattering in a phased array transducer system. A fundamental improvement in the lateral resolution of ultrasonic echo scanners will result if the weight vector of a large phased array transducer can be modified to account for distortion in the propagation medium. Lateral resolution in most tissue is limited to a few mm by wavefront-distortion-induced sound-speed variations. One important wavefront-distortion source is scattering from local speed variations within large and reasonably homogeneous tissue beds such as the liver. Scattering disperses some energy from the beam and perturbs the wavefront, thereby distorting the image and limiting the resolution to the scale of the distortion. Often, such scattering is weak, meaning that most of the energy in the beam is unscattered. The total field at the receiving transducer is the vector sum of the unscattered and scattered fields. In weak scattering the unscattered field is dominant and the resultant field can be treated as the unscattered field plus a perturbation. The net effect is primarily a distorted phasefront, while the amplitude or modulus of the wavefront remains reasonably intact. Refraction and strong scattering effect the wavefront more severely and are less responsive to these algorithms.