  We study the dynamics of a single active Brownian particle (ABP) in two
spatial dimensions. The ABP has an intrinsic time scale $D_R^{-1}$ set by the
rotational diffusion constant $D_R$. We show that, at short-times $t \ll
D_R^{-1}$, the presence of `activness' results in a strongly anisotropic and
non-diffusive dynamics in the $(xy)$ 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.
