The interpolation theorem for a given logical system claims that derivability of a formula \(A \rightarrow B\) implies existence of a formula \(I\) in the language \(L(A)\cap L(B)\) such that \(A\rightarrow I\) and \(I\rightarrow B\) are both derivable. This book is the first volume of a planned two-volume treatise covering the most famous aspects of the work of the two authors in this field. The scope is restricted to propositional and predicate modal and superintuitionistic logics with short asides about related nonclassical systems. Inside this area the coverage is comprehensive. The chapters on algebraic and Kripke models constitute a rigorous streamlined introduction that can be useful in many situations. The complete classification of superintuitionistic propositional systems and modal extensions of S4 and K4 (with respect to interpolation) that constitutes the core of the book is given in full detail. Predicate extensions are covered as well, but here our knowledge is very far from being complete. Model-theoretic aspects are much more prominent than proof-theoretic techniques that appear at the end in connection with applications in computer science. Interpolation is presented in parallel with the Beth definability property: If \(A(P)\& A(P')\rightarrow( P\iff P')\) is derivable, than \(P\) is explicitly definable: \(A(P)\rightarrow (P\iff \phi)\) for a formula \(\phi\) not containing \(P\). The index to the book is very useful and the list of references covers 19 pages. This book is likely to become a definitive treatment of the topic in its title.