Armed officers were called after reports of a woman being assaulted at a property in Tracey Court, off Hibbert Street, Luton, at about 13:30 GMT.
The 24-year-old man was shot by an officer during the incident.
He was taken to hospital and died at 14:50 GMT. The force has referred the incident to the Independent Police Complaints Commission.
A spokeswoman for Bedfordshire Police said the man, who was described as white and British, was initially treated at the scene by officers and paramedics.
His family has been informed.
Chief constable Jon Boutcher said: "First and foremost our thoughts are very much with the family of the man involved in this tragic incident.
"The Independent Police Complaints Commission (IPCC) has been informed, as is standard protocol when someone has died following police contact, and as such it would be inappropriate for us to comment any further at this stage."
The police watchdog confirmed it had sent investigators to the scene and also to the hospital.
In a statement. the IPCC said: "The investigation is in its early stages and no further information is available at this time."
A man has died after he was shot by a police officer, Bedfordshire Police has said.