One problem related to transmission not engage has been reported for the 2001 Volvo S60. The most recently reported issues are listed below. Please also check out the statistics and reliability analysis of the 2001 Volvo S60 based on all problems reported for the 2001 S60.
Stopped to drop off passenger, when attempted to pull away, transmission failed to engage. Turned vehicle off to check it. On restart engaged fine. Approx 1/2 mile down the road, transmission failed again while vehicle was in motion. Coasted to the side of the road, turned vehicle off for a few minutes, on restart transmission once again engaged for me to go a few hundred feet to safety. Vehicle was towed to dealership which diagnosed the transmission with 'internal faults' which would require a complete replacement of the transmission. The vehicle in question has never had any warnings that the transmission was failing, nor had it ever hesitated to engage prior to it's complete failure. The vehicle is 4 1/2 years old with only 32k miles.