Many nations switched from the Julian to the Gregorian, calendar, but they did it at different times. The United States switched in 1752. When there is doubt, we specify dates as "OS" or "NS," for "old style" (Julian) or "new style" (Gregorian).

When I was a kid we used to celebrate "George Washington's Birthday" on February 22nd. (George Washington was the first president of the US and led the US army in the American Revolution). So everybody "knows" that he was born on February 22nd. I have been able to win bets by knowing that because he was born in 1732, before the calendar change, at the time he was born his birthdate was actually February 11th.

It's just a sort of ceremonial thing on some monuments, it isn't really used at all outside them, but in Memorial Hall at Harvard there is some inscription that gives a date both in years AD and in years "since the founding of the City of Cambridge," phrased in Latin. And I've seen monuments in Boston that give a date in years since the founding of Boston. Both of these are in Latin, and are imitating the practice in ancient Rome of dating things "AUC," ab urbe condita, "years since the founding of the city" (i.e. Rome). In the 1800s in the US, we liked all of our big public monuments and buildings to imitate those of ancient Rome.

@Michael (מיכאל), I'm afraid you're blocked or something because I can't see your comment... (which is a shame because I was hoping somebody would explain how it all works in Israel).

It's year 5779 in Israel. 

Come to Israel and welcome to the future)))