And, was everyone to beware of them, or just Caesar?
The short answers are: the conclusion of the the new year's festivities and, really, mostly just Caesar...and some of his enemies.
Across the millenium or so of Roman history, the significance and observation of the Ides of March varied. It's believed that the earliest Roman calendars were lunar, and the Ides of March would have been the first full moon of the new year.
In general terms, the Ides of any month woud be the 13th or 15th, depending upon the length of the month -- essentially, the midpoint. Held sacred to Jupiter, the Ides of each month were observed by the monthly sacrifice of a sheep. March's Ides coincided with the feast of the goddess of the year (Anna Perenna), to wrap up the two-week celebration of the new year. Eventually, by the later part of the Roman Empire, the week of the Ides accumulated various other religious days commemorating mythical entities of Rome's past.
Now, as for Caesar's assassination in 44 BC, the way Plutarch tells it, a seer warned him that some harm would befall him by the Ides. As the story goes, Caesar passed the seer that day on the way to the Curia of Pompey, the location of his doom, and teased him that the Ides had come and he was fine. The seer purportedly responded, "Aye, Caesar; but not gone." This record is probably what gave rise to Shakespeare's "Beware" warning. Note that Shakespeare moved the scene of the assassination to the capitol.
Four years later, Julius Caeser's successor Octavian (later Augustus) executed 300 prisoners of the civil war that followed the assassination -- on the Ides of March, at the site of an altar that had recently been dedicated to Julius.
Beware the Ides of March, indeed.
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