We investigate higher order versions of the concentration of measure phenomenon by means of logarithmic Sobolev inequalities. The functions we consider are non-Lipschitz but have bounded derivatives (or differences) of some higher order $d$. This results in exponential tail bounds which are no longer sub-Gaussian but, for large $t$, typically decay like $\exp(-t^{2/d})$. Our main tool is the entropy method, more precisely $L^p$ norm inequalities derived from log-Sobolev inequalities. A special focus is put on so-called finite spin systems, or functions of weakly dependent random variables, for which we derive log-Sobolev inequalities under suitable (Dobrushin-type) conditions. This is joint work with S. Bobkov, F. Götze and A. Sinulis.