Combinatorial covering properties as Rothberger’s, Hurewicz’s and Menger’s are procedures for generating a cover of a given topological space from a sequence of covers of this space.

We present the most celebrated such properties together with the most important examples in a classical countable case. We also explore how these notions and examples extend to the uncountable context, where the initial sequence of covers has length $\kappa$ for some uncountable cardinal $\kappa$. In this generalized setting, we replace the classical Baire space ${\omega }^{\omega }$ with the generalized Baire space ${\kappa }^{\kappa }$. This is joint work with Piotr Szewczak and Lyubomyr Zdomskyy.