This is the third edition of the author's book from 1977 and 1985. The present, again extended, contents reads as follows: 0. Boolean and Heyting Algeras: The Essentials, 1. Boolean-valued Models of Set Theory: First Steps, 2. Forcing and Some Independence Proofs, 3. Group Actions on V(B) and the Independence of Axiom of Choice, 4. Generic Ultrafilters and Transitive Models of ZFC, 5. Cardinal Collapsing, Boolean Isomorphism, and Applications to the Theory of Boolean Algebras, 6. Iterated Boolean Extensions, Martin's Axiom, and Souslin's Hypothesis, 7. Boolean-valued Analysis, 8. Intiuitionistic Set Theory and Heyting-Algebra-Valued Models, Appendix. Boolean and Heyting Algebra-Valued Models as Categories. The book serves as an introductory course to forcing and completely omits the rapid development in this area during the last 25 years. Instead, it pays some attention to intuitionistic set theory and presents a brief outline of category theory in order to show Boolean and Heyting algebra-valued models as toposes.