There is a 67 page review from last year, <A HREF="http://arxiv.org/abs/1404.5886">Log-concavity and strong log-concavity: a review</A>, A. Saumard, J.A. Wellner (2014):

> We review and formulate results concerning log-concavity and
> strong-log-concavity in both discrete and continuous settings. We show
> how preservation of log-concavity and strongly log-concavity on $\mathbb{R}$
> under convolution follows from a fundamental monotonicity result of
> Efron (1969). We provide a new proof of Efron’s theorem using the
> recent asymmetric Brascamp-Lieb inequality due to Otto and Menz
> (2013). Along the way we review connections between log-concavity and
> other areas of mathematics and statistics, including concentration of
> measure, log-Sobolev inequalities, convex geometry, MCMC algorithms,
> Laplace approximations, and machine learning.

... and here are a whole bunch of older references:

 1. <A HREF="http://link.springer.com/content/pdf/10.1007/BF02243398.pdf">A universal generator for discrete log-concave distributions,</A>
    W Hörmann (1994).
    
 2.   <A HREF="http://epub.wu.ac.at/1756/1/document.pdf">A simple universal generator for continuous and discrete univariate T-concave distributions,</A> J. Leydold (2001).

 3. <A HREF="http://arxiv.org/abs/math/0502548">Preservation of log-concavity on summation</A>, O. Johnson, C. Goldschmidt (2005).

 3.   <A HREF="http://arxiv.org/abs/math/0603647">Log-concavity and the maximum entropy property of the Poisson
    distribution,</A> O. Johnson (2006).

 4.   <A HREF="http://arxiv.org/abs/0805.4112">On the entropy and log-concavity of compound Poisson measures,</A> O. Johnson, I. Kontoyiannis, M.
    Madiman (2008).
    
 5.   <A HREF="http://arxiv.org/abs/0912.0581">Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete
    compound Poisson measures,</A> O. Johnson, I. Kontoyiannis, M.
    Madiman (2009).
        
 6.   <A HREF="https://www.stat.washington.edu/research/reports/2012/tr600.pdf">Strong log-concavity is preserved by convolution,</A> J.A. Wellner
    (2010).

 7.  <A HREF="http://arxiv.org/abs/1107.3904">Asymptotics of the discrete log‐concave maximum likelihood estimator
    and related applications,</A> F. Balabdaoui, H. Jankowski, K. Rufibach, M. Pavlides, (2011).