The 33 year old was officially unveiled today at Turf Moor on a three-and-a-half year deal.
He replaces Brian Laws as Clarets boss and has been joined by his assistant Jason Tindall.
Howe said: "It feels fantastic for me. It's an honour and a privilege to be named Burnley manager and we're looking forward to the challenge ahead.
"It was a difficult week last week. On Tuesday I was approached by two clubs but I turned them down to stay at Bournemouth. Then on Thursday Burnley came in and it was just too good an opportunity to turn down. It was a great challenge for Jason and me, and our families, and we felt we had to take it.
"You just have to look around the stadium and on Saturday [against QPR] I could see the desire of the supporters. There's a huge history to the club, the board have been fantastic with us, and there's a really talented squad as well. Put that in the mix and it's a really good opportunity."