  POMDPs are useful models for systems where the true underlying state is not
known completely to an outside observer; the outside observer incompletely
knows the true state of the system, and observes a noisy version of the true
system state. When the number of system states is large in a POMDP that often
necessitates the use of approximation methods to obtain near optimal solutions
for control. This survey is centered around the origins, theory, and
approximations of finite-state POMDPs. In order to understand POMDPs, it is
required to have an understanding of finite-state Markov Decision Processes
(MDPs) in \autoref{mdp} and Hidden Markov Models (HMMs) in \autoref{hmm}. For
this background theory, I provide only essential details on MDPs and HMMs and
leave longer expositions to textbook treatments before diving into the main
topics of POMDPs. Once the required background is covered, the POMDP is
introduced in \autoref{pomdp}. The origins of the POMDP are explained in the
classical papers section \autoref{classical}. Once the high computational
requirements are understood from the exact methodological point of view, the
main approximation methods are surveyed in \autoref{approximations}. Then, I
end the survey with some new research directions in \autoref{conclusion}.
