This survey aims to discuss some aspects of the semantics of boolean valued models, also bringing forward their presentation in terms of presheaves. In particular, we separate and characterize (also in topological and sheaf-theoretic terms) the fullness property and the mixing property which are the key properties governing the semantics of boolean valued models, we present a duality connecting boolean valued models to presheaves on boolean algebras, and we outline that this duality identifies boolean valued models with the mixing property with sheaves for the dense Grothendieck topology. Consequently, we give a topological description of the sheafification process for presheaves on boolean algebras, and we see its implications in the theory of boolean valued models. Finally, using this topological presentation of the sheafification, we propose a definition of boolean ultrapowers in terms of spaces of continuous functions, which generalizes one proposed by Mansfield in the seventies.