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.