  Recently, additive combinatorics has blossomed into a vibrant area in
mathematical sciences. But it seems to be a difficult area to define - perhaps
because of a blend of ideas and techniques from several seemingly unrelated
contexts which are used there. One might say that additive combinatorics is a
branch of mathematics concerning the study of combinatorial properties of
algebraic objects, for instance, Abelian groups, rings, or fields. This
emerging field has seen tremendous advances over the last few years, and has
recently become a focus of attention among both mathematicians and computer
scientists. This fascinating area has been enriched by its formidable links to
combinatorics, number theory, harmonic analysis, ergodic theory, and some other
branches; all deeply cross-fertilize each other, holding great promise for all
of them! In this exposition, we attempt to provide an overview of some
breakthroughs in this field, together with a number of seminal applications to
sundry parts of mathematics and some other disciplines, with emphasis on
computer science and cryptography.
