A viscosity modifier for lubricating oils or an additive composition for lubricating oils, which is excellent in oil-thickening properties and can provide lubricating oil compositions excellent in low-temperature characteristics and handleability at low temperatures; and lubricating oil compositions excellent in low-temperature characteristics and handleability at low temperatures. The viscosity modifier comprises an ethylene/α-olefin copolymer (B) which comprises (i) a structural unit derived from ethylene, (ii) a structural unit derived from an α-olefin having 3 to 19 carbon atoms, and (iii) a structural unit derived from a higher α-olefin having 4 to 20 carbon atoms whose carbon number is by one or more larger than that of the α-olefin having the unit (ii) which has the following properties of (1) and (2): (1) the contents of units (i), (ii) and (iii) are 25-49 mol %, 15-55 mol %, and 9-40 mol % respectively (with the proviso that the total of the units (i), (ii), and (iii) is 100 mol %) and (2) the intrinsic viscosity [η] is 0.5 to 5.0 dl/g.