Abstract: Indirect measurements were present in the statistical structure of QM from its first development by von Neumann, and their basic mathematically theory was later described by Kraus. Under the name quantum non-demolition measurements, the theory became influential both experimentally and as a toy model for measurement process in quantum mechanics. A recent example of the former is the photon counting experiment of Haroche. The main aim of the lecture would be to introduce the theory and to describe its richness in connections to several mathematical structures: C* algebras, the estimation theory, equivalence of ensembles, martingales, etc. The talk is based on a joint work with M. Ballesteros, J. Frohlich and B. Schubnel.