The Waring problem asks whether for each positive integer n, every positive integer is the sum of a bounded number of nth powers. Although Hilbert solved it more than 100 years ago, it remains the focus of much current research. I will discuss some questions in group theory, especially algebraic group theory, inspired by it. In particular, I will present work with Aner Shalev and Pham Tiep on Waring-type word maps on finite simple groups and work with Dong Quan Ngoc Nguyen on the unipotent Waring problem. The emphasis throughout will be on unsolved problems.