The log-concavity of a sequence is a much studied concept in combinatorics with surprising links to many other mathematical fields. In this talk we discuss the stronger but much less studied notion of m-fold log-concavity which has recently recieved some attention after Boros and Moll conjectured that a "remarkable" sequence encountered in the integration of an inverse cubic is infinitely log-concave. In particular, we report on a recent result of Branden which implies infinite log-concavity of the binomial coefficients and other new developments. Examples and conjectures are promised. A PDF of the talk is available here.