I will discuss some number theoretic background to explain classical reciprocity laws of number theory. This will be an introduction to non-abelian reciprocity which generalizes Artin reciprocity map. I will do this in terms of complex representations of local Galois groups which are to parametrize irreducible admissible representations of GL(n,F), where F is a local field. This is called the Local Langlands correspondence (LLC) for GL(n) which has been established by Harris-Taylor, Henniart and Scholze in various forms. I will mention recent results in matching of different arithmetic (Artin) root numbers and L-functions with analytic ones through (LLC). A global Langlands correspondence for number fields remains illusive and very much out of reach. But one can still discuss one of its major consequences, Langlands functoriality principle, for which there has been some striking progress in the past twenty years