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We consider a generic system operating under non-equilibrium conditions. Explicitly, we consider an inertial classical Brownian particle dwelling a periodic structure with a spatially broken reflection symmetry. The particle is coupled to a bath at the temperature $T$ and is driven by an unbiased time-periodic force. In the asymptotic long time regime the particle operates as a Brownian motor exhibiting finite directed transport although no net biasing force acts on the system. Here we review and interpret in further detail recent own research on the peculiar transport behaviour for this setup. The main focus is put on those different emerging Brownian diffusion anomalies. Particularly, within the transient, time-dependent domain the particle is able to exhibit anomalous diffusive motion which eventually crosses over into normal diffusion only in the asymptotic long-time limit. In the latter limit this normal diffusion coefficient may even show a non-monotonic temperature dependence, meaning that it is not monotonically increasing with increasing temperature, but may exhibit instead an extended, intermediate minimum before growing again with increasing temperature.