Active feature acquisition (AFA) studies how to sequentially acquire features for each data instance to trade off predictive performance against acquisition cost. This survey offers the first unified treatment of AFA via an explicit partially observable Markov decision process (POMDP) formulation. We place this formulation in the broader literature on optimal information acquisition and, more specifically, in a family of structured POMDPs (for example, information-gathering and sensing POMDPs) whose assumptions and algorithmic tools directly apply to AFA. This connection provides a common language for comparing problem settings and methods, and it highlights where AFA can leverage established results in structured POMDP planning and approximation. Building on this perspective, we present an up-to-date taxonomy of AFA methods that (roughly) mirrors standard approaches to solving POMDPs: (i) embedded cost-aware predictors (notably cost-sensitive decision trees and ensembles), (ii) model-based methods that plan using learned probabilistic components, (iii) model-free methods that learn acquisition policies from simulated episodes, and (iv) hybrid methods that combine the strengths of model-based and model-free approaches. We argue that this POMDP-centric view clarifies connections among existing methods and motivates more principled algorithm design. Since much prior work is heuristic and lacks formal guarantees, we also outline routes to guarantees by connecting AFA to adaptive stochastic optimization. We conclude by highlighting open challenges and promising directions for future research.