The spaces of analytic functions that we will present were introduced by Louis de Branges and James Rovnyak in 1966. The original purpose was to make a connection with operator theory on Hilbert space, and to develop a certain model theory in order to obtain functional representations for classes of contractions. From this point of view, these spaces appear as generalizations of the model spaces (backward-shift invariant subspaces of the Hardy-Hilbert space). But considered as spaces of analytic functions on the unit disc per se, they have acquired in the last decades a life of their own. They have been shown to exhibit various interesting and sometimes unexpected properties, and have become an interesting and often challenging object of study.

The present short course will try to provide the audience with some of the flavour of de Branges-Rovnyak spaces. Developing the theory normally requires delving into technical details that may be complicated to the novice; we will try to avoid this path as far as possible and rather illustrate general properties by examples, occasionally giving short proofs. An important topic is the study of certain operators on the spaces, making thus the link with model theory. Other topics considered will be:

- Regularity of functions in the space

- Integral representations

- Multipliers

- Relations to other function spaces

- Carleson measures

- Sequences of reproducing kernels

The basic references for de Branges-Rovnyak spaces are the original short but informative book of Sarason [1] and the comprehensive monography of Fricain and Mashreghi [2].

[1] Donald Sarason, Sub-Hardy Hilbert spaces in the unit disk, John Wiley & Sons Inc., New York, 1994.

[2] Emmanuel Fricain and Javad Mashreghi, The theory of H(b) spaces, Vol. 1 and 2, Cambridge University Press, Cambridge, 2016.