From the Preamble: According to {\it P. Lindström} [“First-order predicate logic with generalized quantifiers”, Theoria 32, 186-195 (1966)] a quantifier is a functor which assigns to each non-empty domain a relation among relations which is closed under isomorphisms. A simple instance of this notion is given by the quantifier ‘more than half of the’, which for each domain $E$ gives the relation between sets $A,B \subseteq E$ defined by: $$|\{a\in A: a\in B\}|> |\{a\in A: a\not\in B\}|.$$ In the present collection of articles the authors investigate several aspects of such quantifiers, also of quantifiers with relational arguments. This introduction presents some basic insights and techniques of quantification theory. After a brief history, we pay attention to application of the theory in linguistics, and then to its more logical features. The linguistic topics include: denotational constraints, behaviour in certain linguistic contexts, and polyadic forms of quantification. On the logical side, we discuss metaproperties of weak and of ‘real’ quantifier logics. In particular, we concentrate on the tableau method for weak quantifier logics, and on decidability results.