We suggest the name “approximation theorems” for a new kind of theorems which are strong versions of preservation theorems. A typical preservation theorem has the form: A sentence φ is preserved by the relation R (i.e. and imply ) iff there exists φ* ϵ Φ such that ⊧ φ ↔ φ* where R is a relation between structures and Φ is a class of sentences (depending, of course, on R ). Usually, Φ is described in syntactical terms and it is easy to see that every element of it is, indeed, preserved by R . A typical approximation theorem has the form: For every sentence Φ there is a sentence Φ* ϵ Φ (the “approximation” of Φ) such that, for all sentences δ which are preserved under R , (a) if ⊧δ → φ then ⊧δ → φ* and (b) if ⊧φ → δ then ⊧φ* → δ. An approximation theorem obviously implies the corresponding preservation theorem. The first approximation theorem was proved by Vaught in [14] (see Corollary 2.3 below). That paper inspired the present one. Vaught's result is stated in topological terms. It says that for each Borel set B there is an invariant Borel set B * explicitly defined by an L ω1ω sentence, such that B − ⊆ B * ⊆ B + where B − ( B + ) is the largest (smallest) invariant set included in (containing) B .