UEFA said the move, approved by its executive committee, would allow smaller countries a slice of the action as they would not be able to stage an unwieldy, 24-team tournament on their own.
UEFA said it planned to use a mix of major cities and medium and smaller ones for the tournament which will mark the competition's 60th anniversary.
"The main point is really to give more cities, more countries the possibility to host a celebration European championship," UEFA secretary general Gianni Infantino said.
"Instead of a party in one country, you will have a party all over Europe."
The idea was first aired by UEFA president Michel Platini at the end of Euro 2012. Three of the last four European Championships have been staged in two countries although the tournament was traditionally held in one.
The announcement threw up a range of questions such as how many cities would host games, whether all host countries would qualify and play at home, and how much travelling would be done.
Infantino said that decisions would be made over next few months.
"Today, everything is open," he said.
He said the bidding process would start by March with a final decision on the host cities by spring 2014.
Infantino said that financial considerations played a major part in the decision.
"The fact that the Euro moves to 24 teams bears additional burdens on the host nations, the requirements are becoming bigger and bigger," he said.
"One of the purposes of this decision is to help countries who are maybe not sure today whether they should build or not a national stadium, to give them this impetus.
"It will be a lot easier from a financial perspective for all the countries. If you need to build airports or 10 stadiums in a country - this would be rather easy because it would be one stadium per host city."
Infantino said the many new logistical challenges would be addressed in the next few months, including the number of host cities.
"The only thing decided is that the concept is changing," he said. "How many cities is to be decided and determined."
Other issues to be addressed include whether to play groups in one or two venues close to each other, or whether to force teams to play their group games in different parts of the continent.
"If groups are going to be in different countries, then ideally they should not be too far away," he said. "We have to look into that and take a decision which makes sense.
He also accepted that with so many host countries, some of them would not qualify.
"You can't have 15 hosts automatically qualified. This would have to be thought about and looked at," he said.
"Another question is will hosts who qualified play at home... It will probably be difficult to understand for the fans of a country which is host if they do not play a match at home.
"Probably, you have a certain number of teams at home against a certain number of teams who play away, this may be an issue. But there's also the question of if it's actually an advantage to play at home."
Infantino said the change was a one-off decision to celebrate the 60th anniversary. "It is a decision only about 2020, it is not going forward forever."
How Euro 2020 could be hosted:
- Sports & Recreation
- Gianni Infantino