The survey examines the current state of the speed gradient method, developed in the 1970-80s for synthesizing control and adaptation algorithms in nonlinear systems, as well as numerous applications of the method to solving scientific and engineering problems. Brief information about speed gradient algorithms and conditions of their applicability, optimality, and passivity are given. Applications of the method to problems of adaptive control and identification, nonlinear control, energy and nonlinear oscillation control, and control in network, multiagent, and distributed systems are discussed. Applications of the method to control of technical systems and to problems in physics, biology, and ecology are presented. Modern modifications and generalizations of the method are provided, including non-Euclidean speed gradient algorithms based on Lyapunov-Bregman functions.