The actor - or well, actress - currently topping that betting list, however, is Olivia Colman. The British actress has accumulated a pretty lengthy resume, including Hot Fuzz, Fleabag, and Broadchurch.
Colman's Broadchurch costar and former Time Lord himself, David Tennant, has weighed in on the rumors of her casting, and appears to be all for it.
"Olivia would clearly be a magnificent choice," Tennant told the British website The World of One.
This rousing endorsement, combined with Broadchurch creator Chris Chibnall becoming Doctor Who's executive producer in the series' twelfth season, makes Colman's casting seem more and more likely.
Tennant echoes this, saying that this change in showrunners - after Steven Moffat's lengthy involvement with the series - could be perfect for making the transition to Colman. "If you have the right people telling the right stories," Tennant says, "then it's absolutely a possibility."
Some fans have been pushing for a female Doctor for quite some time, with names such as Agent Carter's Hayley Atwell and Preacher's Ruth Negga being suggested for the Thirteenth Doctor. But with Colman's casting being a serious possibility, there is a chance it could help push Doctor Who into a whole new era.
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Doctor Who returns for Season 10 in April 2017.