A short exact sequence of Banach spaces is a diagram of Banach spaces and operators $0 \to Y \to X \to Z \to 0$ in which the kernel of every operator agrees with the image of the preceding. Since short exact sequences are made of operators, \emph{and} we all know that there are compact operators between Banach spaces, one may ask: when, if at all, should a short exact sequence be called “compact”?

The purpose of this talk is to show that this notion of compact exact sequence is meaningful and fruitful. We recall some of the basic notions of short exact sequences in Banach spaces, together with some of its applications, and adapt them to the setting of compact operators between Banach spaces. In particular, we present some results on the extension of compact operators and develop the first steps of a theory of compact exact sequences. This is a joint work with F. Cabello Sánchez, J. M. F. Castillo and N. Trejo-Arroyo.