The Brief Amy Klobuchar has made it official, and is running for governor of Minnesota. Klobuchar, a Democrat, is currently in her fourth term in the U.S. Senate. U.S. Sen. Amy Klobuchar has made it official, and is running for governor of Minnesota after Gov. Klobuchar is running for MN governorWhat we know:Last week, Klobuchar took the first steps toward running for governor by filing preliminary paperwork, and on Thursday morning made the official announcement with a more than 4-minute video posted to social media. The backstory:Klobuchar, 65, is in her fourth Senate term, first running in 2006 after a career as a Hennepin County prosecutor.