Cosmic rays are highly energetic charged particles of astrophysical origin. While most of the cosmic-ray flux measured at Earth is made up of protons, a small fraction consists of positrons. The local cosmic-ray positron flux is shaped by several powerful astrophysical processes. While the positron flux at lower energies (< 20 GeV) is explained by secondary production in interactions of other cosmic rays, the origin of the unexpectedly large flux at high energies, referred to as the positron excess, is not clearly understood. A prominent explanation for the high-energy flux are pulsars, rapidly rotating neutron stars that convert part of their rotational energy into highly-energetic electron-positron pairs. Additional contributions may come from dark matter particles that annihilate into final states containing positrons, which would appear as a sharply peaked feature in the positron flux. In the first part of this thesis, measurements of the local cosmic-ray positron flux are used to derive constraints on such dark matter models, improving on previous constraints.

After their injection and acceleration to high energies by various types of sources, cosmic-ray positrons rapidly lose energy as they propagate through the Galaxy, due to interactions with the Galactic magnetic fields (synchrotron losses) and interactions with the photons of the interstellar radiation fields (inverse-Compton scattering losses). A precise understanding of these energy loss processes is vital to accurately model and predict the contribution of individual sources to the locally measured positron flux. The second part of this thesis presents improved calculations of these energy loss processes. In particular, it treats inverse-Compton losses precisely as a stochastic processes rather than using the continuous energy loss approximation adopted across the literature. The improved model leads to significant changes in the expected features of the positron flux: While individual pulsars produce much smoother features than previously thought, the dark matter signals instead become enhanced. This makes it possible to distinguish the pulsar and dark matter features clearly -- and, importantly, leaves dark matter as the only known astrophysical source that is capable of producing sharp features in the positron flux. This establishes the local cosmic-ray positron flux as one of the most promising probes for dark matter signals.