To be eligible, passengers can either wear their military uniform or show the bus driver their service ID card on Remembrance Sunday (10th November) and Armistice Day (Monday 11th November).
It is available on all First Bus services apart from the Airport Flyer.
The free travel offer is being made available to make it as easy as possible for customers to get to and from remembrance services and events.
Rob Pymm, Commercial Director for First West of England and Cymru, said: “Armistice Day and Remembrance Sunday are important and solemn reminders of the sacrifices made by so many in our armed forces, and we want to do something to mark these occasions.
“Across First Bus, a number of our colleagues are veterans, and we’re very proud of their efforts and sacrifice. This is our way of saying thank you to them, as well as honouring the men and women who choose to serve and protect the freedoms we now enjoy.
“We want to make bus travel to and from remembrance events as easy as possible for people who are attending, and I am pleased that our business can play our part in supporting and remembering those in military service past and present.”