Although there is a particular beauty in the statement that the space of bounded linear operators in a Banach space is itself a Banach space, the norm in this space is more often than not too strong to encompass more delicate convergence theorems of contemporary mathematics. Strong convergence is a notion that is more suitable for such purposes. We exemplify this by studying two classical theorems: Bernstein’s approximation of continuous functions by polynomials and the theorem of Fej\’er on convergence of Fourier series. In both cases the operators involved converge strongly but not in the operator norm. Before doing that, however, we discuss the theorem of Banach and Steinhaus. This result ensures in particular that in Banach spaces strong convergence of bounded linear operators implies boundedness of the limit operator. The chapter also covers the famous Poisson approximation to the binomial, the only example of a limit theorem of probability known to the author that can in fact be stated in the framework of norm convergence of operators.