Abstract: Random walk and Brownian motion among static obstacles have been the subject of extensive research during the last couple of decades. Of particular interest have been the survival probability as well as the path behavior of the motion conditioned on survival. However, comparatively less attention has been given to scenarios involving moving traps. We survey some of the results obtained in the setting of a Poisson cloud of moving traps during the last decade and discuss a recently established functional central limit theorem in dimensions 6 and higher.