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 Cantor space $2^\omega$ and the classical Baire space $\omega^\omega$ with the $\kappa$-Cantor space $2^\kappa$ and the $\kappa$-Baire space $\kappa^\kappa$, respectively. This is joint work with Piotr Szewczak and Lyubomyr Zdomskyy.