1. To chase a person or animal all around an area or from place to place. I thought a trip to the museum would be fun, but I just spent the whole time running around after the kids. The first job my uncle gave me on the farm was to run around after the chickens so we could give them their vaccinations.

2. To pursue someone or something with continual persistence in order to obtain or ensure something. As the project manager, most of my day is spent running around after all the various pieces other people are working on. I've had to run around after these builders for weeks to make sure the work gets finished on time.