This chapter contains classical results about Baire-type conditions (Baire-like, b-Baire-like, CS-barrelled, s-barrelled) on tvs. We include applications to closed graph theorems and \(C(X)\) -spaces. We also provide the first proof in book form of a remarkable result of Saxon (extending earlier results of Arias de Reyna and Valdivia), which states that, under Martin’s axiom, every lcs containing a dense hyperplane contains a dense non-Baire hyperplane. This part also contains analytic characterizations of certain completely regular Hausdorff spaces X. For example, we show that X is pseudocompact, is Warner bounded, or \(C_c(X)\) is a \((df)\) -space if and only if for each sequence \((\mu _n)_n\) in the dual \(C_c(X)^{\prime }\) there exists a sequence \((t_n)_n \subset (0,1]\) such that \((t_n \mu _n)_n\) is weakly bounded, strongly bounded, or equicontinuous, respectively. These characterizations help us produce a \((df)\) -space \(C_c(X)\) that is not a \((DF)\) -space, solving a basic and long-standing open question.