Here's a look at six of the leading candidates to replace the departing Frenchman:
1) Angel Maria Villar
The 66-year-old Spaniard is probably the early favourite, although it is a tight market, simply because he has held senior positions at FIFA and UEFA since 1992.
A former midfielder for Athletic Bilbao and Spain, Villar studied law while still playing and helped found the Spanish players' union. He then ran the Spanish FA from 1988 to 2008, and would later lead Spain and Portugal's bid for the 2018 World Cup. But that role means he is compromised in the eyes of many, and he is under investigation for possible corruption during that campaign.
He has also been criticised for not speaking out, during or after, FIFA's long crisis. He may be too tainted by his association with the ancien regime.
2) Theodore Theodoridis
Another front-runner, he is relatively unknown outside of Greece and the corridors of UEFA's headquarters. A former board member of the Greek FA, the 50-year-old joined UEFA in 2008 as a director and became Gianni Infantino's deputy as general secretary in 2010.
Since March, he has been doing Infantino's old job on an interim basis. He is on this list largely because of his reputation for solid, understated efficiency but he lacks pizzazz. He also brings some baggage of his own as his father Savvas is the vice-president of Olympiacos, the Greek giants currently embroiled in a match-fixing scandal.
3) Giancarlo Abete
A UEFA vice-president since 2011, the Italian has held an extraordinary number of positions in Italian business, politics and sport. The son of a leading businessman, the 65-year-old spent more than a decade in politics before becoming Italian FA president in 2007.
He has also served on numerous sub-committees at UEFA and FIFA, so he knows how the system works and has so far managed to stay out of trouble. But Abete is another short on wow factor and it is unclear how badly he really wants the top job.
4) David Gill
The former Manchester United chief executive will be many people's first choice for this role. The problem is, he is many people's first choice for other jobs in football. A trained accountant, Gill made his mark in business before joining Manchester United as finance director in 1997.
He rose through the ranks at Old Trafford and forged a winning partnership with Sir Alex Ferguson. Some United fans regret his departure in 2013 almost as much as the brilliant Scot's. He has also held senior positions at the FA and European Club Association.
The question mark surrounding a Gill candidacy, is will the 58-year-old Englishman want an executive position in Europe when he has a crucial role to play as a reform-minded FIFA vice-president?
5) Zbigniew Boniek
More of an outsider than the others, the 60-year-old Pole is the president of his national FA and hugely popular in his native land. In fact, there are many similarities between Boniek and Platini: both were brilliant players for the same Juventus team, both starred for their countries and then managed their national sides.
Boniek would certainly look the part and carry lots of support in eastern Europe and Italy. But he has never held a position at UEFA and FIFA, which may spook the voters slightly.
6) Michael Van Praag
The Dutchman is the last on this list but is perhaps the best candidate of all. A successful businessman who also refereed in the amateur leagues in Holland, van Praag followed his father's footsteps by becoming chairman of Ajax from 1989 to 2003.
He then took over as president of the Dutch FA in 2008 and has served as a UEFA vice-president since 2015. He will be 69 in September, when the UEFA presidential election is likely to take place, so he is not young. But he earned a lot of credit for standing against Sepp Blatter in last year's FIFA race. European football could do much worse than handing the reins to van Praag for a few years of quiet diplomacy.