<p>Controllability has become a crucial aspect of trustworthy machine learning, enabling learners to meet predefined targets and adapt dynamically at test time without requiring retraining as the targets shift. We provide a formal definition of controllable learning (CL), and discuss its applications in information retrieval (IR) where information needs are often complex and dynamic. The survey categorizes CL according to what is controllable (e.g., multiple objectives, user portrait, scenario adaptation), who controls (users or platforms), how control is implemented (e.g., rule-based method, Pareto optimization, hypernetwork, and others), 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.</p>