  Matter at extreme temperatures and pressures -- commonly known as warm dense
matter (WDM) in the literature -- is ubiquitous throughout our Universe and
occurs in a number of astrophysical objects such as giant planet interiors and
brown dwarfs. Moreover, WDM is very important for technological applications
such as inertial confinement fusion, and is realized in the laboratory using
different techniques. A particularly important property for the understanding
of WDM is given by its electronic density response to an external perturbation.
Such response properties are routinely probed in x-ray Thomson scattering
(XRTS) experiments, and, in addition, are central for the theoretical
description of WDM. In this work, we give an overview of a number of recent
developments in this field. To this end, we summarize the relevant theoretical
background, covering the regime of linear-response theory as well as nonlinear
effects, the fully dynamic response and its static, time-independent limit, and
the connection between density response properties and imaginary-time
correlation functions (ITCF). In addition, we introduce the most important
numerical simulation techniques including ab initio path integral Monte Carlo
(PIMC) simulations and different thermal density functional theory (DFT)
approaches. From a practical perspective, we present a variety of simulation
results for different density response properties, covering the archetypal
model of the uniform electron gas and realistic WDM systems such as hydrogen.
Moreover, we show how the concept of ITCFs can be used to infer the temperature
from XRTS measurements of arbitrarily complex systems without the need for any
models or approximations. Finally, we outline a strategy for future
developments based on the close interplay between simulations and experiments.
