You can do both depending on the style of bag it is. A “handbag” is on that has handles or not that can only be carried. A bag that has either a shoulder strap or a “crossbody” strap can be worn.
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One theory suggests that carrying a bag in the left hand originated from the practice of carrying a sword on the left side of the body. This allowed the sword to be easily drawn with the right hand while still holding the bag with the left hand.
Technically, any piece of luggage that you “carry on” to an airplane is a carry-on bag. Most airlines allow one piece of carry-on luggage or “hand baggage” that can fit in the overhead bin, plus a “personal item” (a smaller purse, computer bag, diaper bag, small backpack, etc.
You can't carry on a large suitcase even if you want to, so sometimes the amount of stuff you plan to take on a trip makes the carry-on versus checked bag decision for you. If you're going on a two-day business trip, for example, it's unlikely you'll need more than a small rollaboard and a laptop bag.