We first investigate the properties of consistence, strongness and semimodularity, each of which may be viewed as a generalization of modularity. Chapters 2 and 3 present the decomposition theory of lattices. Here we characterize modularity in terms of the Kurosh-Ore replacement property. Next, we study c-decompositions of elements in lattices (the notion of c-decomposition is introduced as a generalization of those of join and direct decompositions). We find a common generalization of the Kurosh-Ore Theorem and the Schmidt-Ore Theorem for arbitrary modular lattices, solving a problem of G. Grätzer. The decomposition theory for lattices enables us to develop a structure theory for algebras. Some kinds of representations of algebras are studied. We consider weak direct representations of a universal algebra. The existence of such representations is considered. Finally, we introduce the notion of an ⟨ℒ,ψ⟩-product of algebras. We give sufficient conditions for an algebra to be isomorphic to an ⟨ℒ,ψ⟩-product with simple factors and with directly indecomposable factors. Some applications to subdirect, full subdirect and weak direct products are indicated.