Dark matter constitutes six times more of the mass in the Universe than ordinary baryonic matter. The halos formed by dark matter are fundamental to the formation of structure, yet the properties of these halos are not well understood theoretically or observationally.

In the first part of this thesis, I review the evidence for the existence of dark matter and its role in cosmology and astrophysics, and I discuss the particle nature of dark matter and how that nature can be identified. Then, I summarize the predicted properties of collisionless dark matter halos from fundamental theory and from numerical simulations of the formation of structure. In particular, I focus on the possibility of an anisotropic velocity dispersion tensor where the typical velocity in one direction can be different from that in another direction. This anisotropy demonstrates the fundamental difference between collisionless dark matter systems and collisional gases.

In the second part of the thesis, I investigate how the properties of halos can be constrained observationally, and what these constraints imply about dark matter. First, I analyze how accurately the velocity anisotropy parameter can be measured in a dark matter detector which is sensitive to the direction of the measured dark matter particles. The anisotropy parameter turns out to be measurable, but I find that a very large number of events are necessary to reject isotropy at high statistical significance.

Second, using x-ray observations of a sample of galaxy clusters, I carry out a first measurement of the radially-varying velocity anisotropy profile of dark matter. The measured profile is found to be non-zero and radially increasing, which is in good agreement with the predictions of numerical simulations. The measurement implies that dark matter is in fact effectively collisionless in clusters and it sets an upper limit on the dark matter self-scattering cross section.

Finally, I again use a sample of clusters observed in x-ray to constrain the mass distribution of dark matter in the halos of the clusters. I compare the measured mass profiles with a number of parametrized mass models in a detailed statistical analysis. I find moderate evidence that the observed mass profiles are not strictly universal but require a shape parameter to be fitted to each halo.