Logarithmic correlators or responses in non-relativistic analogues of conformal invariance

Recent developments on emergence of logarithmic terms in correlators or response functions of models which exhibit dynamical symmetries analogous to conformal invariance in not necessarily relativistic systems are reviewed. The main examples of these are logarithmic Schr\"odinger-invariance and logarithmic conformal Galilean invariance. Some applications of these ideas to statistical physics are described.