abstract: We study the dynamics of a single active Brownian particle (ABP) in two spatial dimensions. The ABP has an intrinsic time scale set by the rotational diffusion constant. We show that, at short-times, the presence of activity results in a strongly anisotropic and non-diffusive dynamics in the x-y plane. We compute exactly the marginal distributions of the x and y position coordinates along with the radial distribution, which are all shown to be non-Brownian. In addition, we show that, at early times, the ABP has anomalous first-passage properties, characterized by non-Brownian exponents. We also study the dynamics of ABP in a two-dimensional harmonic trap and show that the stationary position distribution in the trap exhibits two different behaviours: a Gaussian peak at the origin in the strongly passive limit and a delocalised ring away from the origin in the opposite strongly active limit. The predicted stationary behaviours in these limits are in agreement with recent experimental observations.