It is shown that a valuation v of a field K is ''complete by stages'' (Ribenboim) or ''perfect'' (Krull) if and only if its residual fields are complete, where the residual fields of v are the fields \(A_ P/P\), where A is the valuation ring of v, P a nonmaximal prime ideal of A, furnished with the topology defined by the valuation induced by v, and short proofs of known theorems concerning this property are given. Extending a classical theorem of \textit{A. Ostrowski} [J. Reine Angew. Math. 143, 255- 284 (1913)], the author shows that if v' is an extension of a Henselian valuation v of a field K to a separable algebraic extension L and if L is a Baire space for the topology defined by v', then \([L:K]<+\infty\) and K is closed in L. Finally, equivalent conditions are given for a valuation v of K to be half Henselian, that is, to have precisely two extensions to the algebraic closure of K. For example, v is half Henselian if and only if there is a (possibly improper) Henselian valuation u of K whose valuation ring strictly contains that of v such that the residue field k of u is real-closed and the valuation induced on k by v is not Henselian. Consequently, a field K admits a half Henselian valuation if and only if it admits a Henselian valuation whose residue field is real-closed.