Just a note of caution if you are going to use a variable frequency drive (VFD) on an existing motor it should be manufactured with at least Class F insulation and be rated for inverter power source. Because of the PWM (Pulse Width Modulated) waveform there is residual heating and most motors will go to 1.0 SF from 1.15 SF. There can also be issues relating to lead lengths in excess of 100 feet or more, output filtering may be required, ie. 3-5% reactor. Grounding sources become an issue as well and ungrounded or questionable ground sources can cause bearing fretting to occur, called common mode noise.
Don't be discouraged I just caution any piece of electronic equipment on a system can contribute to additional issues and need to be addressed or considered for a VFD on an existing motor. As many have stated the vast use is to provide an efficient power source for variable output such as centrifugal loads, pumps/fans. They can also limit torque and current to control starting and stopping of high torque loads, this also needs to be considered when using existing motors. Long acceleration/deceleration may require motors to be oversized or blower cooled. You can also use the same long deceleration load features to common DC bus a load on power applications to power additional VFDs for a safe shutdown, system dependent.
Speed matching, master/follower applications for material handling are opportunities as well. PID, flow control, speed matching, electronic overload, shear pin, skip frequency, communications to a network/SCADA system are also excellent ideas.
This said they are not always easy to place in a system as a drop in replacement. With large loads, overhauling loads, extended start stop times, critical applications, etc. it is best to define the control system desired and determine best options. To vary speeds I would always want to know the application, pump/fan, blower, conveyor, etc. Any special requirements for starting or stopping. Control or communication requirements. Distance from VFD to motor, any speeds or speed range required.