Newton–Kantorovich method and its global convergence

In 1948, L. V. Kantorovich extended the Newton method for solving nonlinear equations to functional spaces. This event cannot be overestimated: the Newton–Kantorovich method became a powerful tool in numerical analysis as well as in pure mathematics. We address basic ideas of the method in the historical perspective and focus on some recent applications and extensions of the method and some approaches to overcoming its local nature.