  The second law of thermodynamics dictates that heat simultaneously flows from
the hot to cold bath on average. To go beyond this picture, a range of works in
the past decade show that, other than the average dynamical heat flux
determined by instantaneous thermal bias, a non-trivial flux contribution of
intrinsic geometric origin is generally present in temporally driven systems.
This additional heat flux provides a free lunch for the pumped heat and could
even drive heat against the bias. We review here the emergence and development
of this so called ``geometric heat pump'', originating from the topological
geometric phase effect, and cover various quantum and classical transport
systems with different internal dynamics. The generalization from the adiabatic
to the non-adiabatic regime and the application of control theory are also
discussed. Then, we briefly discuss the symmetry restriction on the heat pump
effect, such as duality, supersymmetry and time-reversal symmetry. Finally, we
examine open problems concerning the geometric heat pump process and elucidate
their prospective significance in devising thermal machines with high
performance.
