In 1937, Stefan Kaczmarz proposed a simple method, called the Kaczmarz algorithm, to solve iteratively systems of linear equations Ax = b in Euclidean spaces. This procedure employs cyclic orthogonal projections onto the hyperplanes associated with such a system. In the case of a nonsingular matrix A, Kaczmarz showed that his method guarantees convergence to the solution of Ax = b. The Kaczmarz algorithm was rediscovered in 1948 and became an important tool in medical engineering. We briefly discuss generalizations of this method and its ramifications, including applications in computer tomography, image processing and contemporary harmonic analysis.