The Conditional Logical Framework LFK is a variant of the Harper-Honsell-Plotkin's Edinburgh Logical Framemork LF. It features a generalized form of λ-abstraction where β-reductions fire under the condition that the argument satisfies a logical predicate. The key idea is that the type system memorizes under what conditions and where reductions have yet to fire. Different notions of β-reductions corresponding to different predicates can be combined in LFK. The framework LFK subsumes, by simple instantiation, LF (in fact, it is also a sub-system of LF!), as well as a large class of new generalized conditional λ-calculi. These are appropriate to deal smoothly with the side-conditions of both Hilbert and Natural Deduction presentations of Modal Logics. We investigate and characterize the metatheoretical properties of the calculus underpinning LFK, such as subject reduction, confluence, strong normalization.