According to the conventions of meteorology, today, June 1, is the official first day of summer, at least for those of us in north of the tropics. Similarly, autumn starts September 1, winter December 1, and spring March 1.
Of course, lots of people will tell you that the "official first day of summer" doesn't happen until June 21, the date of the June solstice. That's when the Earth reaches the point in its orbit where the North Pole is tilted the maximum amount toward the Sun. Actually, the solstice is an instant in time, which this year is about 1639 (4:39 pm) June 21 in London, or 1139 June 21 in New York, but 0239 (2:39 am) June 22 in Sydney. The exact time and date vary from year to year based mainly on where we are in the leap year cycle.
So where do people get the idea that the equinoxes and solstices mark the "official" beginnings of seasons? It may possibly go back to the old convention, observed from the Middle Ages into the 1500s in much of Western Europe, of treating the spring equinox as the first day of the new year.
But to complicate matters, it wasn't actually the spring (or "vernal") equinox as such that served as New Year's Day but rather its then-traditional date of March 25, which had been the usual date of the equinox in the early years of the Roman Empire. (The December solstice, and hence the feast of the Unconquered Sun, Sol Invictus, was similarly around December 25, whence we get the date of Christmas.)
By the 1500s the relatively simple Julian calendar (with one leap year in every four, no exceptions) was about two weeks off from the march of the seasons, so to put it back in sync with the seasons when the Gregorian calendar was introduced (which skips over three leap years in every four centuries), the date also jumped ahead by 10 days. This wasn't quite enough days to match up with the old Roman conventions, which is why our equinoxes and solstices take place typically on the 21st or 22nd of the month in question rather than the 25th.
Besides that history, it's also true that early March isn't quite so spring-like as later in the month, and early December is similarly not so wintry, so treating the solstices and equinoxes as the beginnings of seasons isn't so unreasonable.
There's noting "official" about it, though.