Controllable learning (CL) emerges as a critical component in trustworthy machine learning, ensuring that learners meet predefined targets and can adaptively adjust without retraining according to the changes in those targets. We provide a formal definition of CL, and discuss its applications in information retrieval (IR) where information needs are often complex and dynamic. The survey categorizes CL according to who controls (users or platforms), what is controllable (e.g., retrieval objectives, users' historical behaviors, controllable environmental adaptation), how control is implemented (e.g., rule-based method, Pareto optimization, Hypernetwork), and where to implement control (e.g.,pre-processing, in-processing, post-processing methods). Then, we identify challenges faced by CL across training, evaluation, task setting, and deployment in online environments. Additionally, we outline promising directions for CL in theoretical analysis, efficient computation, empowering large language models, application scenarios and evaluation frameworks in IR.