  A majority of methods from dynamical systems analysis, especially those in
applied settings, rely on Poincar\'e's geometric picture that focuses on
"dynamics of states". While this picture has fueled our field for a century, it
has shown difficulties in handling high-dimensional, ill-described, and
uncertain systems, which are more and more common in engineered systems design
and analysis of "big data" measurements.
  This overview article presents an alternative framework for dynamical
systems, based on the "dynamics of observables" picture. The central object is
the Koopman operator: an infinite-dimensional, linear operator that is
nonetheless capable of capturing the full nonlinear dynamics. The first goal of
this paper is to make it clear how methods that appeared in different papers
and contexts all relate to each other through spectral properties of the
Koopman operator. The second goal is to present these methods in a concise
manner in an effort to make the framework accessible to researchers who would
like to apply them, but also, expand and improve them. Finally, we aim to
provide a road map through the literature where each of the topics was
described in detail.
  We describe three main concepts: Koopman mode analysis, Koopman
eigenquotients, and continuous indicators of ergodicity. For each concept we
provide a summary of theoretical concepts required to define and study them,
numerical methods that have been developed for their analysis, and, when
possible, applications that made use of them.
  The Koopman framework is showing potential for crossing over from academic
and theoretical use to industrial practice. Therefore, the paper highlights its
strengths, in applied and numerical contexts. Additionally, we point out areas
where an additional research push is needed before the approach is adopted as
an off-the-shelf framework for analysis and design.
