Anderson will retire as chief executive in June and will complete his term as vice chairman.
Anderson said, "I've always wanted to leave the organisation at the right interlude: when I saw a new leader ready to take the organisation to a new level, higher than I could take it myself.”
Dunn said: "Because of the work Brad has led, Best Buy has earned the opportunity to play a unique role in the lives of people - helping them unlock the promise of technology. We also believe the value we can bring to people is timeless and universal, applicable literally around the world."
Dunn, Best Buy’s president and chief operating officer, has been at the company for 23 years and has worked his way up from the shop floor since he joined in 1985.
Dunn added: "I'm honoured and excited to lead the next chapter in this company's amazing story."
A decision about who will replace Dunn in his current role has not yet been announced.
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