In this PhD thesis we investigate various aspects of particle dark matter. The proper identification of dark matter developed during the second half of the twentieth century to become one of the biggest endeavours in modern physics and astronomy. Although observations currently favour the explanation that dark matter consists of a new form of particle, no experimental search has yet provided unequivocal evidence of such a particle. Of particular importance in this thesis is the field of indirect detection of dark matter, where one searches for the particles emerging from annihilations of dark matter particles out in the Universe. Specifically, we consider dark matter annihilations in the centre of the Sun. As the Sun moves through the galaxy, some dark matter particles scatter in the Sun and lose enough energy to become bound to the Sun. They settle in the solar core and begin to annihilate, which leads to an annihilation signal from the solar direction. The thesis is built on novel research consisting of three papers and a monograph-type chapter. In the first paper we calculate the flux of high energy neutrinos coming from cosmic ray cascades in the solar atmosphere and investigate the role it plays as a background in solar dark matter searches. In the second paper we consider dark matter annihilating into longlived mediators in the Sun, which leads to interesting new detection possibilities. A third paper explores more generally the fluxes of secondary particles from dark matter annihilations that are searched for in indirect detection. We look at the effects of changing the Monte Carlo event generator that generates the fluxes and of having polarized final states in the annihilations. Finally, we consider in a monograph-type chapter the production of dark matter in the early Universe through the freeze-out mechanism, looking at effects of higher order corrections in the calculation of the relic abundance in the minimal supersymmetric standard model.