Model theory on domains in {\mathbb C}^n

Take a domain in {\mathbb C}^n and a positive definite kernel k on it which is holomorphic in the first variable and anti-holomorphic in the second. Under some natural conditions on k, dilation of a class of commuting bounded tuples of operators has been obtained by Ambrosie, Englis and Muller. However, is there a functional model? In general, the answer is no. In this talk, we shall start from the very basics of the theory which originated in connection with function theory on the disk and then gradually proceed to the state of the art situation.