Properties of elementary operators, especially on algebras of operators, have been vigorously studied over the past decades. It emerges that the situation of the Calkin algebra is full of surprises. In this setting, an elementary operator with dense range is surjective, and injectivity entails boundedness below. In the case of Hilbert space, positivity implies complete positivity, and the norm and the cb-norm of every elementary operator coincide. We present an overview on some results of this

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... lavour, in particular on recent extensions of the latter two results to antiliminal-by-abelian C*-algebras (obtained by Archbold, Somerset, and the author), and strong rigidity properties of the norm of elementary operators on Calkin algebras due to Saksman and Tylli.