Reciprocity is a general, fundamental principle governing various physical systems, which ensures that the transfer function—the transmission of a physical quantity, say light intensity—between any two points in space is identical, regardless of geometrical or material asymmetries. Breaking this transmission symmetry offers enhanced control over signal transport, isolation and source protection1,2,3,4,5,6. So far, devices that break reciprocity (and therefore show non-reciprocity) have been mostly considered in dynamic systems involving electromagnetic, acoustic and mechanical wave propagation associated with fields varying in space and time. Here we show that it is possible to break reciprocity in static systems, realizing mechanical metamaterials7,8,9,10,11,12,13,14,15,16that exhibit vastly different output displacements under excitation from different sides, as well as one-way displacement amplification. This is achieved by combining large nonlinearities with suitable geometrical asymmetries and/or topological features. In addition to extending non-reciprocity and isolation to statics, our work sheds light on energy propagation in nonlinear materials with asymmetric crystalline structures and topological properties. We anticipate that breaking reciprocity will open avenues for energy absorption, conversion and harvesting, soft robotics, prosthetics and optomechanics.