For nearly a century, more mass has been measured in galaxies than is contained in the luminous stars and gas. Through continual advances in observations and theory, it has become clear that the dark matter in galaxies is not comprised of known astronomical objects or baryonic matter, and that identification of it is certain to reveal a profound connection between astrophysics, cosmology, and fundamental physics. The best explanation for dark matter is that it is in the form of a yet undiscovered particle of nature, with experiments now gaining sensitivity to the most well-motivated particle dark matter candidates. In this article, I review measurements of dark matter in the Milky Way and its satellite galaxies and the status of Galactic searches for particle dark matter using a combination of terrestrial and space-based astroparticle detectors, and large scale astronomical surveys. I review the limits on the dark matter annihilation and scattering cross sections that can be extracted from both astroparticle experiments and astronomical observations, and explore the theoretical implications of these limits. I discuss methods to measure the properties of particle dark matter using future experiments, and conclude by highlighting the exciting potential for dark matter searches during the next decade, and beyond.