It is shown that in systems with time-dependent and/or spatially nonuniform temperature T(t, r), the diffusion (Brownian motion) is anomalous (AD). A few examples of simple arrangements, easy for experimental realization, are discussed in detail. Proposed measurements will enable also the observation of transitions from normal to anomalous diffusion. New effects are predicted: (i) zero-mean oscillations of T(t) accelerate AD (pumping effect), (ii) the combination of temporal and spatial variations of temperature may lead to superballistic AD, (iii) various forms of T(t, r) may result in non-algebraic AD including exponential diffusion. One can expect similar effects in inflationary systems with time-dependent metric and in expanding/contracting gases.