Abstract: In this talk we will discuss some results concerning the zero sets of real analytic functions on open sets in $\mathbb{R}^n$. We will consider the related notion of analytic uniqueness sequences and, as an application, we will show that the zero set of every non-constant real analytic function on a domain has always empty interior with respect to the fine topology (which strictly contains the Euclidean one). Further, we will see that for a certain category of sets $E$ (containing the finely open sets), a function is real analytic on some open neighbourhood of $E$ if and only if it is real analytic ''at each point'' of $E$. (Joint work with Andre Boivin and Paul Gauthier.)