A Stone algebra is a distributive lattice with pseudocomplement that satisfies the Stone identity \(a^*\sqcup a^{**}=1\). The author determines all quantifier elimination classes of Stone algebras and all classes of Stone algebras that admit positive quantifier elimination (i.e. every positive existential formula is equivalent to a positive quantifier-free formula). The author reproves some model theoretic results on Stone algebras using his results about quantifier elimination and gives necessary conditions for quantifier elimination for pseudocomplemented distributive lattices.