  A stunning range of substructures in the dust of protoplanetary disks is
routinely observed across a range of wavelengths. These gaps, rings and spirals
are highly indicative of a population of unseen planets, hinting at the
possibility of current observational facilities being able to capture
planet-formation in action. Over the last decade, our understanding of the
influence of a young planet on the dynamical structure of its parental disk has
progressed significantly, revealing a host of potentially observable features
which would betray the presence of a deeply embedded planet. In concert, recent
observations have shown that subtle perturbations in the kinematic structure of
protoplanetary disks are found in multiple sources, potentially the
characteristic disturbances associated with embedded planets. In this work, we
review the theoretical background of planet-disk interactions, focusing on the
kinematical features, and the current methodologies used to observe these
interactions in spatially and spectrally resolved observations. We discuss the
potential pit falls of such kinematical detections of planets, providing
best-practices for imaging and analysing interferometric data, along with a set
of criteria to use as a benchmark for any claimed detection of embedded
planets. We finish with a discussion on the current state of simulations in
regard to planet-disk interactions, highlighting areas of particular interest
and future directions which will provide the most significant impact in our
search for embedded planets. This work is the culmination of the 'Visualizing
the Kinematics of Planet Formation' workshop, held in October 2019 at the
Center for Computational Astrophysics at the Flatiron Institute in New York
City.
