Medium- and high-entropy alloys (M/HEAs) mix several principal elements with near-equiatomic composition and represent a model-shift strategy for designing previously unknown materials in metallurgy^1-8, catalysis^9-14 and other fields^15-18. One of the core hypotheses of M/HEAs is lattice distortion^5,19,20, which has been investigated by different numerical and experimental techniques^21-26. However, determining the three-dimensional (3D) lattice distortion in M/HEAs remains a challenge. Moreover, the presumed random elemental mixing in M/HEAs has been questioned by X-ray and neutron studies²⁷, atomistic simulations^28-30, energy dispersive spectroscopy^31,32 and electron diffraction^33,34, which suggest the existence of local chemical order in M/HEAs. However, direct experimental observation of the 3D local chemical order has been difficult because energy dispersive spectroscopy integrates the composition of atomic columns along the zone axes^7,32,34 and diffuse electron reflections may originate from planar defects instead of local chemical order³⁵. Here we determine the 3D atomic positions of M/HEA nanoparticles using atomic electron tomography³⁶ and quantitatively characterize the local lattice distortion, strain tensor, twin boundaries, dislocation cores and chemical short-range order (CSRO). We find that the high-entropy alloys have larger local lattice distortion and more heterogeneous strain than the medium-entropy alloys and that strain is correlated to CSRO. We also observe CSRO-mediated twinning in the medium-entropy alloys, that is, twinning occurs in energetically unfavoured CSRO regions but not in energetically favoured CSRO ones, which represents, to our knowledge, the first experimental observation of correlating local chemical order with structural defects in any material. We expect that this work will not only expand our fundamental understanding of this important class of materials but also provide the foundation for tailoring M/HEA properties through engineering lattice distortion and local chemical order.