Hungary's most notable showing at an international tournament is the strange and ultimately sad story of the 1954 World Cup. Hungary's Golden Team, the Magical Magyars — Ferenc Puskas, Nandor Hidegkuti, Sandor Kocsis and the rest — reached the final after a run of 32 internationals without defeat. Having thumped their opponents West Germany in the group stage, they were heavy favorites to repeat the trick, and went 2-0 up early … before everything fell apart.
In the driving rain Germany pulled the game back to 2-2 by halftime, frustrated the Hungarian attack for most the second half, then took the lead with six minutes to play. Four minutes later Puskas had the ball in the German net, only for the linesman to flag, perhaps incorrectly. And two minutes after that, the "Miracle of Bern" was complete. Subsequent investigations have suggested that the German side may have been injected with amphetamines before the game, though too much time has passed for certainty either way; regardless, Hungary's best shot at the World Cup had gone.
When it comes to the Euros, however, we need to look a little later in history. The Golden Team broke up in 1956; the European Nations' Cup didn't begin until 1960, and Hungary's previous two appearances came in 1964 and 1972. Back then, the finals were a four-team competition. In the two preceding years, European nations played two-legged qualification ties against one another, before the last four nations standing would come together to settle the final stages. So Hungary made their way to Spain in June 1964 by overcoming Wales in November 1962, East Germany almost a year later in October 1963, and then finally France in April 1964.
The star of that post-Golden Team side was Florian Albert, one of those irritatingly complete footballers who scored goals in industrial quantities — 256 in 351 games for his only club, Ferencvaros, as well as 31 in 75 for his country — while still managing to accrue adjectives such as "elegant" and "graceful." He remains the only Hungarian to have won the Ballon d'Or, in 1967, but three years earlier he was unable to find the net in Spain. The Hungarians took the hosts to extra time in the semifinal, then were taken to extra time themselves in the third-fourth playoff. Goals from Ferenc Bene and Dezso Novak saw them past Denmark for the bronze medal.
Eight years later, things didn't go quite so well. The group stage had been introduced by this point, and after qualifying past Bulgaria, France and Norway, Hungary needed a last-minute winner against Romania to secure their place in the final four. Once there, however, they were rolled over by the Soviet Union and then hosts Belgium; Albert again failed to score, and their only goal was a consolation penalty in the third-fourth playoff.
It's worth noting that while this team doesn't have quite the historical reputation of the Golden Team, they weren't just confined to a couple of appearances at the nascent Euros. The 1960s side — the Silver Team, perhaps — took bronze in the 1960 Olympics and then gold the following two Olympiads. They reached the quarterfinals of the 1962 and 1966 World Cups; the latter campaign included a victory over Brazil in the group stage, the holders' first loss at a World Cup since 1954. That too, by happy coincidence, had come at the hands and feet of Hungary.
Since then, it's been a barren period for Hungarian international football. Until this year they'd never again qualified for the final stages of the Euros, and though they made it to the World Cup in 1978, 1982 and 1986, they couldn't get out of their groups. But while this current group doesn't have anybody to rival Puskas or Albert, they had the good fortune to be drawn into a remarkably open-looking group. Then there's the fact that third place may be enough to get into the knockout stages. In short, a bit of luck early on and Hungary may find itself just three successful penalty shootouts away from a chance to avenge the ghosts of 1954.