First, we review the classical theory of Asplund and weak Asplund properties of Banach spaces. Equivalent, sufficient, and necessary conditions for them are listed. In particular, the role of differentiability of the norm is shown. Stability under various gymnastics is considered. Open questions naturally raised are mentioned. Deeper structural results for Asplund property are shown via projectional resolutions of identity (PRI) and a recent mutant of it, projectional skeletons. Here, the so-called rich families of separable subspaces arise naturally. An application in (sub-)differentiability theory is sketched then. Finally, we raise a question whether it is possible to study both weak Asplund and Asplund properties in locally convex spaces. Nowadays, this is a quite vibrant area. Sometimes we can imitate the technology from Banach spaces, but not always. Some of the obtained results are a bit surprising. Definitely, it makes sense to go beyond Banach spaces. Just to get a taste: Mazur's theorem that separable Banach spaces are weak Asplund can be extended to separable Baire t.l.s.; the non-Banach spaces $C_k(\mathbb{Q})$ and $\mathbb{R}^\aleph$, with $\aleph$ any cardinal, are Asplund; and given any Banach space $X$, then $(X,w)$ and $(X^*,w^*)$ are Asplund.