Piezoresistivity is an electromechanical effect characterized by the reversible change in the electrical resistivity with strain. It is useful for electrical-resistance-based strain/stress sensing. The resistivity can be the volumetric, interfacial or surface resistivity, though the volumetric resistivity is most meaningful scientifically. Because the irreversible resistivity change (due to damage or an irreversible microstructural change) adds to the reversible change that occurs at lower strains, the inclusion of the irreversible effect makes the piezoresistivity appear stronger than the inherent effect. This paper focuses on the inherent piezoresistivity that occurs without irreversible resistivity changes. The effect is described by the gage factor (GF), which is defined as the fractional change in resistance per unit strain. The GF can be positive or negative. Strong piezoresistivity involves the magnitude of the fractional change in resistivity much exceeding the strain magnitude. The reversible effect of strain on the electrical connectivity is the primary piezoresistivity mechanism. Giant piezoresistivity is characterized by GF >= 500. This critical review with 209 references covers the theory, mechanisms, methodology and status of piezoresistivity, and provides the first review of the emerging field of giant piezoresistivity. Piezoresistivity is exhibited by electrically conductive materials, particularly metals, carbons and composite materials with conductive fillers and nonconductive matrices. They include functional and structural materials. Piezoresistivity enables structural materials to be self-sensing. Unfortunately, GF was incorrectly or unreliably reported in a substantial fraction of the publications, due to the pitfalls systematically presented here. The most common pitfall involves using the two-probe method for the resistance measurement.