Methods of performing an anastomosis. A ring is installed at an incision or other orifice in a vessel or other organ, for use in performing an anastomosis. To install the ring, tines are advanced against an anvil to cause them to grab tissue around the orifice and curl radially. The anvil is then retracted to fold the tines so their curled ends move radially outward. A sleeve may be advanced to fold or bend the tines as their curled ends move further radially outward. The folding, curling, and further bending of the tines evert tissue near the orifice edges to expose the vessel's or organs' intima.