  The Koopman operator allows for handling nonlinear systems through a
(globally) linear representation. In general, the operator is
infinite-dimensional - necessitating finite approximations - for which there is
no overarching framework. Although there are principled ways of learning such
finite approximations, they are in many instances overlooked in favor of, often
ill-posed and unstructured methods. Also, Koopman operator theory has
long-standing connections to known system-theoretic and dynamical system
notions that are not universally recognized. Given the former and latter
realities, this work aims to bridge the gap between various concepts regarding
both theory and tractable realizations. Firstly, we review data-driven
representations (both unstructured and structured) for Koopman operator
dynamical models, categorizing various existing methodologies and highlighting
their differences. Furthermore, we provide concise insight into the paradigm's
relation to system-theoretic notions and analyze the prospect of using the
paradigm for modeling control systems. Additionally, we outline the current
challenges and comment on future perspectives.
