A classical result in complex analysis is that a meromorphic function on the complex plane is a ratio of two entire functions. A very simple example of a meromorphic function is a rational function. In the noncommutative setting rational functions are a more complicated object. The free skew field (the skew field of all noncommutative rational functions) was constructed by Amitsur and by Cohn. A very concrete way to look at these objects is via realizations. Realizations allow us to deal with linear objects and a single inversion for the price of dealing with (perhaps very big) matrices and vectors. In this talk, I will discuss two constructions of noncommutative meromorphic functions, one arising from realizations and one from a purely algebraic construction. The first one gives interesting results even in the single variable case whereas the second one provides a set of new examples of topological semi-free ideal rings. If time will permit, I will discuss an application to free probability.

The talk is based on joint work with Meric Augat and Rob Martin.