This talk addresses advances in the local theory of noncommutative functions, which branches in two directions. First we will see that the ring of noncommutative functions analytic about a scalar point admit a universal skew field of fractions, whose elements are called meromorphic germs. We also obtain a local-global rank principle for matrices of analytic noncommutative functions. On the other hand, if $Y$ is a semisimple (non-scalar) point, then there exist nilpotent analytic noncommutative functions about $Y$. Nevertheless, the ring of germs about $Y$ can be described as the completion of the free algebra with respect to the vanishing ideal at $Y$. This is a consequence of our second main result, a free Hermite interpolation theorem: if $f$ is a noncommutative function, then for any finite set of semisimple points and a natural number $L$ there exists a noncommutative polynomial that agrees with $f$ at the chosen points up to differentials of order $L$.

This is joint work with Igor Klep and Victor Vinnikov.