Let f be a transcendental meromorphic function. A p-periodic component U of the Fatou set of f is called a Baker domain if the boundary of U contains a point z such that the iterates of f^p tend to z on U but f^p(z) is not defined.

Baker domains of transcendental entire functions are simply connected (by a result of Baker). This is not true in general for transcendental meromorphic functions. In my talk I will discuss the relationship between the multiple connectivity of Baker domain and the existence of weakly repelling fixed points. In particular I will prove a result on the simple connectivity of Baker domains for Newton's method. This is joint work with K. Baranski, N. Fagella and X. Jarque.