In this chapter we pursue the interplay between field-theoretic and model-theoretic concepts already explored in Chaps. 7 and 8 and expand it to algebraic geometry. Our main goal is to offer a concrete illustration of the fact that, in one respect, model theory provides a ‘logical analysis’ of certain algebraic results, framing in an abstract form the conditions or properties that, within specific algebraic settings, coincide with independently studied mathematical content. We certainly do not claim that model theory is to be viewed mainly or primarily as an instrument of logical analysis, but we find it important to emphasise this role, which is both mathematically and philosophically interesting. It is mathematically interesting because it shows that model-theoretic ideas and techniques provide a new outlook on familiar objects and results. It is philosophically interesting because it shows that model-theoretic results govern a reconstruction of mathematical content along novel, distinctively logical, lines. Section 10.1 gives two model-theoretic characterisation of field-theoretic notions. The first is a simple observation to the effect that ‘algebraic element’ in the model-theoretic sense specialises to ‘algebraic element’ in the field-theoretic sense. The second and central result of this section is a beautiful theorem proved by McKenna, McIntyre and Van den Dries, to the effect that the infinite fields with quantifier elimination are exactly the models of \(\textsf {ACF}\) . This theorem provides a purely model-theoretic characterisation of ‘algebraically closed’. In Sect. 10.2 we introduce the rudiments of algebraic geometry. Our central goal here is to obtain a proof of Hilbert’s Nullstellensatz based on the model-completeness of \(\textsf {ACF}\) . In the process we illustrate some natural model-theoretic translations of geometric notions like that of a variety. Section 10.3 concludes our brief exploration of the interplay between model theory and algebraic geometry. This section contains a first introduction to Morley rank and a proof that this notion coincides with that of dimension of a variety when set in the context of algebraic geometry.