Classical Romans reckoned dates in a strange way. To summarize, they counted backwards from certain "special" days in each month. Your date--June 27th--would be written a.d. VI Kal. Jul.; here's why:
Every month has three special days: The Kalends which was always the first day of the month, the Ides which fell on either the 15th (in March, July, October, and May) or the 13th (every other month), and the Nones, which always fell eight days earlier than the Ides (i.e. on the 7th in March, July, October, and May, the 5th in every other month).
To write a date, you first determine the next special day; note this may be the Kalends of the next month if you are in the 2nd half of the current month. Next you count backwards the number of days from that special way to your current date, but there is one slight complication: The Romans counted both the first and last day of a series, so their counting appears to us to include an extra day. For example, we would say January 23rd occurs 5 days before January 28th, but the Romans would actually say January 23rd is the 6th day before January 28th.
The full form of the date is then written ante diem X Kalends/nones/Idus Y where X is the number of days (usually written as a Roman Numeral) and Y is the month containing the special day (in the genitive case). This was often abbreviated a. d. X Kal./Non./Id. Y, with the month Y abbreviated to three letters.
A few small exceptions: If the date happens to be the day of a special day, you simply write e.g. Kal. Nov. = November 1st. If the date is the day before a special day, you write e.g. pridie Kal. Nov. = October 31st. Finally, there is a rare exception for dates in late February during a leap year; I don't remember it off-hand, but I believe they count the 6th day before the March Kalends twice, i.e. a.d. VI Kal. Mar. is usually February 24th, but in a leap year it is Feb. 25th, and Feb 24th is written a.d. bis VI Kal. Mar.. Or something like that...
Some examples:
December 25th - Next special day is the Kalends of January on the 1st. Count back from there to the 25th and you have seven days, but the Romans add and extra day, so this date is written a. d. VIII Kal. Jan.
March 12th - a. d. IV Id. Mar.
April 12th - pridie Id. Apr. (since Ides falls on the 13th in April).
July 2nd - a.d. VI Non. Jul.
Every month has three special days: The Kalends which was always the first day of the month, the Ides which fell on either the 15th (in March, July, October, and May) or the 13th (every other month), and the Nones, which always fell eight days earlier than the Ides (i.e. on the 7th in March, July, October, and May, the 5th in every other month).
To write a date, you first determine the next special day; note this may be the Kalends of the next month if you are in the 2nd half of the current month. Next you count backwards the number of days from that special way to your current date, but there is one slight complication: The Romans counted both the first and last day of a series, so their counting appears to us to include an extra day. For example, we would say January 23rd occurs 5 days before January 28th, but the Romans would actually say January 23rd is the 6th day before January 28th.
The full form of the date is then written ante diem X Kalends/nones/Idus Y where X is the number of days (usually written as a Roman Numeral) and Y is the month containing the special day (in the genitive case). This was often abbreviated a. d. X Kal./Non./Id. Y, with the month Y abbreviated to three letters.
A few small exceptions: If the date happens to be the day of a special day, you simply write e.g. Kal. Nov. = November 1st. If the date is the day before a special day, you write e.g. pridie Kal. Nov. = October 31st. Finally, there is a rare exception for dates in late February during a leap year; I don't remember it off-hand, but I believe they count the 6th day before the March Kalends twice, i.e. a.d. VI Kal. Mar. is usually February 24th, but in a leap year it is Feb. 25th, and Feb 24th is written a.d. bis VI Kal. Mar.. Or something like that...
Some examples:
December 25th - Next special day is the Kalends of January on the 1st. Count back from there to the 25th and you have seven days, but the Romans add and extra day, so this date is written a. d. VIII Kal. Jan.
March 12th - a. d. IV Id. Mar.
April 12th - pridie Id. Apr. (since Ides falls on the 13th in April).
July 2nd - a.d. VI Non. Jul.