Relative to the language of rings \(\mathcal {L}_{r}\) , the algebraically closed fields are exactly the existentially closed fields. The existentially closed ordered fields in the language \(\mathcal {L}_{r<}\) are algebraically known as real closed ordered fields, while their \(\mathcal {L}_{r}\) -reducts are the real closed fields. In this chapter we provide first-order axiomatisations of the above classes of fields, study their key model-theoretic properties and discuss some of their basic applications to real algebraic geometry. Section 11.1 covers essential background on ordered fields. The algebraic characterisation of real closed fields introduced in Sect. 10.2 leads to a formulation of the first-order \(\mathcal {L}_{r}\) -theory \(\textsf {RCF}\) , which axiomatises the class of real closed fields, and of the \(\mathcal {L}_{r<}\) -theory \(\textsf {RCF}_{<}\) , which extends \(\textsf {RCF}\) and axiomatises the class of real closed ordered fields. In Sect. 11.3 we prove quantifier elimination for \(\textsf {RCF}_{<}\) and deduce from it the model-completeness of \(\textsf {RCF}\) . Although the latter theory does not have quantifier elimination, it is, like its extension, complete and decidable. We also obtain an analogue of the main theorem proved in Sect. 10.1 , to the effect that an ordered field has quantifier elimination iff it is a model of \(\textsf {RCF}_{<}\) . Section 11.4 contains some basic applications of the model-theoretic properties of \(\textsf {RCF}_{<}\) , in particular quantifier elimination and o-minimality, to real algebraic geometry.