Abstract: The purpose of this talk is to introduce and advertise the concept of log-concavity and its recent generalizations. A sequence $a_n$ is log-concave if $L(a_n) = a_n^2 - a_{n-1} a_{n+1} \ge 0$, and many important sequences from various fields, such as combinatorics, geometry or number theory, are known or believed to be log-concave. Following Boros and Moll, a sequence $a_n$ is $m$-log-concave if $L^j(a_n) \ge 0$ for all $j = 0, 1, \ldots, m$. A motivating example is the case of binomial coefficients which have been conjectured to be infinitely log-concave. While a recent result of Brändén shows that this is indeed true for rows of Pascal's triangle, the case of columns remains open. Time permitting, we will report on joint work with Luis Medina in which we investigate sequences which are fixed by a power of the log-concavity operator $L$. Surprisingly, we find that sequences fixed by the non-linear operators $L$ and $L^2$ are, in fact, characterized by a linear 4-term recurrence. Plenty of open problems will be mentioned along the way.