A typical polling system consists of a number of queues, attended by a single server in a fixed order. The vast majority of papers on polling systems focus on Poisson arrivals, whereas very few results are available for general arrivals. The current study is the first one presenting simple closed-form approximations for the mean waiting times in polling systems with renewal arrival processes, performing well for all workloads. The approximations are constructed using heavy traffic limits and newly developed light traffic limits. The closed-form approximations may prove to be extremely useful for system design and optimisation in application areas as diverse as telecommunications, maintenance, manufacturing and transportation.