In 1937 Stefan Kaczmarz introduced an iterative method for solving linear systems of equations. Recently, Szwarc and Haller (based on work of Kwapien and Mycielski) characterized convergent Kaczmarz algorithms in infinite-dimensional Hilbert spaces by means of tight frames of effective sequences. The purpose of this talk is to extend some of their results by introducing and characterizing a more general notion of convergence of the algorithm, as well as introducing frames and Riesz bases in this context.