ScotRail has apologised to those customers hit by delays.
The train operator said its “final push” to deliver timetable improvements later this month was the main reason behind the disruption.
Passengers have been advised to check their route online before they travel.
In a message on Twitter, ScotRail said: “Due to train crew shortages some services will be cancelled or altered today. We are sorry for any disruption.”
A later message added: “Thank you for your patience and understanding regarding cancellations due to train crew shortages today. We are sorry if your journey has been affected.”
A number of services have been affected, including some trains between Glasgow and Edinburgh.
Routes linking the central belt with locations such as Aberdeen, Inverness, Dundee and Oban may also be affected.
The new timetable comes into force on Sunday December 9, with ScotRail saying some of the benefits will include faster journeys on the route between Edinburgh Waverley and Glasgow Queen Street via Falkirk High.
The new schedule is understood to require a significant amount of training for drivers and conductors.
A ScotRail spokesman said: “We’re sorry to our customers who have experienced disruption to their journey.
“The majority of the impact is a result of our final push to deliver the December timetable improvements which will bring faster journeys, more seats, and more services for our customers.”