Air carrier Captain reported during gate arrival several service vehicles drove quickly in front of the aircraft; requiring flight crew to brake to avoid collisions.

Narrative

Unsafe company driving - While turning in to park at gate at approximately XA:06L; we observed a baggage cart with its rear wheel across the red line. We stopped and informed ops of the issue. We were 30-35 degrees into the turn; and our nose obstructed the white-line-delineated ground vehicle roadway. The cart continued to back further across the line; and then pulled forward and cleared the safety area. As we released brakes and continued to taxi into the gate; two vehicles swerved away from us (outside the white lines) and zipped past us from left to right. One was a smaller vehicle and the other appeared to be a catering truck which serviced the aircraft next to us. We applied brakes and brought the aircraft to a stop to avoid colliding with the vehicles.ZZZ company driving training and culture is broken. Immediate management attention is required to prevent injuries and equipment damage. Anytime an aircraft has to apply brakes to avoid a collision with a service vehicle; something has gone badly wrong. Company drivers are too complacent around aircraft and/or have inadequate guidance as to when and how to maneuver near aircraft.Management at all levels must immediately get to grips with the nature and severity of this issue. Policies & procedures must be reviewed and; where necessary; amended. Explicit and implicit management pressure to rush to comply must be addressed. Company drivers must be given strict guidelines; and rewarded for staying within them. 'Late and safe' must be rewarded. Unsafe driving must be eliminated.