Holomorphic functions enjoy the following two well-known properties:
- if they vanish in a subset, they must be identically zero,
- they can be approximated by polynomials, under mild conditions on the set of analyticity.
Unique continuation and Runge-type approximation properties share this  same flavour in the setting of solutions to more general partial  differential equations.

In this talk I will review some of these dual properties for both local and nonlocal operators. I will also discuss some applications, including to the context of inverse problems.


