  Low-rank matrix recovery problems arise naturally as mathematical
formulations of various inverse problems, such as matrix completion, blind
deconvolution, and phase retrieval. Over the last two decades, a number of
works have rigorously analyzed the reconstruction performance for such
scenarios, giving rise to a rather general understanding of the potential and
the limitations of low-rank matrix models in sensing problems. In this article,
we compare the two main proof techniques that have been paving the way to a
rigorous analysis, discuss their potential and limitations, and survey their
successful applications. On the one hand, we review approaches based on descent
cone analysis, showing that they often lead to strong guarantees even in the
presence of adversarial noise, but face limitations when it comes to structured
observations. On the other hand, we discuss techniques using approximate dual
certificates and the golfing scheme, which are often better suited to deal with
practical measurement structures, but sometimes lead to weaker guarantees.
Lastly, we review recent progress towards analyzing descent cones also for
structured scenarios -- exploiting the idea of splitting the cones into
multiple parts that are analyzed via different techniques.
