The subconvexity problem aims at providing non-trivial (i.e. subconvex) bounds for central values of automorphic L-functions; the main conjecture in this area is the Generalized Lindelof Hypothesis which itself is a consequence of the Generalized Riemann Hypothesis. This lecture will survey several advances that have been made on this question during the past ten years: these include, the delta-symbol approach of R. Munshi, the Weyl type bounds of I. Petrow and M. Young (both use the Dirichlet L-series representation of the central values) and the works of P. Nelson and A. Venkatesh (which use the automorphic period representations for the central value)