A multi-axis force sensor measures forces and moments occurring in more than one spatial direction. In this way, a single multi-axis sensor can perform what is essentially a three-dimensional measurement of physical quantities. This feature makes multi-axis sensors popular in a wide range of engineering research including automation, machining processes, aerospace, medical applications and civil engineering. Measurement of multi-directional forces and moments is typically achieved using multiple strain-sensitive elements mounted on an elastic structure. Both the sensitive elements and the elastic structure require careful consideration to design a force sensor for accuracy, reliability and robustness. While the development of multi-axis sensors has been considered extensively in the literature over the past seven decades, a collective resource which collates and examines this information does not exist. This review explores multi-axis force sensor developments across a broad range of disciplines. The salient fundamental strain sensing techniques adopted for the strain-sensitive elements reported in the literature are discussed and a critical review of elastic structure designs that have featured in the literature is also presented.