Dynamic arrest in soft matter systems manifests in a variety of arrested states, including gels, attractive and repulsive glasses. Usually, the approach to an arrested state happens through a two-step relaxation of the dynamic auto-correlation functions, where the microscopic, fast relaxation is separated from the structural, slow relaxation by an intermediate plateau often interpreted in terms of the so-called cage effect. However, anomalous dynamics where this picture in not valid is observed in many situations for soft matter systems. In this case, the two-step behavior is replaced by a logarithmic relaxation of the density auto-correlation functions, which is accompanied by a sub-diffusive behavior of the mean-squared displacement. In this talk I will present some examples of anomalous slow dynamics in simulations and experiments, and I will discuss the current theoretical understanding of this behavior.