An active matter system comprises self-propelled units operating outside equilibrium, each with the ability to convert stored or ambient free energy into organized movement. These systems are prevalent in various forms, including self-organizing bio-polymers, the turbulent motion of bacterial systems, and the coordinated flight of bird flocks. In this presentation, I discuss the persistence problem in a spatio-temporally chaotic pattern generated by a dense suspension of bacteria, referred to as active turbulence. Our findings indicate that the persistence time of passive tracers within coherent vortices follows a Weibull probability density in contrast to inertial turbulence. To understand the mixing of passive tracers we propose a measure of complexity, namely the topological entropy, that can be directly computed from the time evolution of a material line made up by these tracers. I will close with a glimpse of our ongoing work on the first passage problem of an active particle in crowded media.