Active feature acquisition (AFA) studies how a predictive system can sequentially choose which feature values to obtain for each instance to balance predictive accuracy against feature acquisition cost. This survey provides the first unified treatment of modern AFA through a partially observable Markov decision process (POMDP) formulation, showing that most existing methods can be understood as different approximations of the same underlying sequential decision problem. The survey proposes an up-to-date taxonomy organizing AFA into four categories: (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.