Not all hybrids do. For instance the Honda IMA "Assist" hybrids have the single motor/generator integrated into the engine flywheel and as such can use any transmission type although the original Insight offered either a 5 speed manual or a cone-and-belt CVT.
Toyota, Nissan, and Ford series/parallel "full" hybrids all use "CVTs" which aren't really CVTs at all but just emulate one. This is so because unlike other types the hybrid and "transmission" functions are one in the same. CVTs use cones (pulleys, sheaves) and a belt or discs and rollers to control the effective ratio. Toyota's HSD and others like it use 2 motor/generators and a PSD to control the ratio. MG1 spinning free is neutral, MG1 turning fast with low torque and MG2 turning slow with high torque is equivalent to low gear, MG1 turning the same RPM as MG2 is direct drive, and MG1 turning slow either forward or backward with high torque and MG2 turning fast with low torque is overdrive.
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