Person search aims to jointly localize and identify a query person from natural, uncropped images, which has been actively studied in the computer vision community over the past few years. In this paper, we delve into the rich context information globally and locally surrounding the target person, which we refer to scene and group context, respectively. Unlike previous works that treat the two types of context individually, we exploit them in a unified globallocal context network (GLCNet) with the intuitive aim of feature enhancement. Specifically, re-ID embeddings and context features are enhanced simultaneously in a multistage fashion, ultimately leading to enhanced, discriminative features for person search. We conduct the experiments on two person search benchmarks (i.e., CUHK-SYSU and PRW) as well as extend our approach to a more challenging setting (i.e., character search on MovieNet). Extensive experimental results demonstrate the consistent improvement of the proposed GLCNet over the state-of-the-art methods on the three datasets. Our source codes, pre-trained models, and the new setting for character search are available at: https://github.com/ZhengPeng7/GLCNet.