Let X be a Tychonoff space. We survey some classic and recent results that characterize the topology or cardinality of X when C p (X) or C k (X) is covered by certain families of sets (sequences, resolutions, closure-preserving coverings, compact coverings ordered by a second countable space) which swallow or not some classes of sets (compact sets, functionally bounded sets, pointwise bounded sets) in C (X). 1. Preliminaries Unless otherwise stated, X will stand for an infinite Tychonoff space. We denote by C p (X) the linear space C(X) of real-valued continuous functions on X equipped with the pointwise topology τ p. The topological dual of C p (X) is denoted by L(X), or by L p (X) when provided with the weak* topology. We denote by C k (X) the space C(X) equipped with the compact-open topology τ k. A family {A α : α ∈ N N } of subsets of a set X is a resolution for X if it covers X and verifies that A α ⊆ A β for α ≤ β. A family of bounded sets in a locally convex space E that swallows the bounded sets is called a fundamental family of bounded sets. Definitions not included in this paper can be found in [6, 18, 49]. 2. Countable coverings for C p (X) The following folklore result can be found in [49, Proposition 9.18]. Velichko's theorem can be found in [1, I.2.1 Theorem] or in [49, Theorem 9.12]. Theorem 2.1. The space C p (X) admits a fundamental sequence of pointwise bounded sets if and only if X is finite. Theorem 2.2 (Velichko). The space C p (X) is covered by a sequence of compact sets if and only if X is finite. Next theorem extends Velichko's result to relatively countably compact sets. Theorem 2.3 (Tkachuk-Shakhmatov [75]). C p (X) is covered by a sequence of relatively countably compact sets if and only if X is finite