This volume contains the proceedings of the international colloquium organized by the Tata Institute of Fundamental Research in January 2016, one of a series of colloquia going back to 1956. The talks at the colloquium covered a wide spectrum of mathematics, ranging over algebraic geometry, topology, algebraic K-theory and number theory. Algebraic theory, homotopy theory and topological K-theory formed important sub-streams in this colloquium. Several branches of K-theory, like algebraic cycles, triangulated categories of motives, motivic cohomology, motivic homotopy theory, Chow groups of varieties, Euler class theory, equivariant K-theory as well as classical K-theory have developed considerably in recent years, giving rise to newer directions to the subject as well as proving results of