5th edition allows any character to pick up two weapons and wield them simultaneously, at least under specific circumstances.

Any character can wield two light one-handed weapons in each hand. Using their once-a-turn bonus action they can make an attack with the off-hand weapon during their round. The off-hand weapon doesn't get your static stat bonus to its damage. And for light weapons in 5e this can be about 50% of the weapon's average per-round damage.

Two classes, the fighter and the ranger, can chose to take a class feature that give the stat damage to the off-hand attack. In addition, if the DM allows them, there is a feat that removes the 'light weapon' requirement.

This seems pretty powerful, but several classes have reasons not to dual-wield.
--All the spell casters generally need to keep a hand free for material or somatic components.
--The monk has a mostly superior version built into their class.
--Barbarians and Paladins will either want a two-handed weapon for superior average damage, or to go 'sword and board' to increase their AC.

Many Rogues, oddly enough, have other things to do with their bonus action. The have abilities that make them the most agile of all the classes, if not the fastest for many levels, and a rogue who hangs out in melee combat to get an off-hand attack is probably going to end up getting killed. (In 5e rogues get one sneak attack per turn, period. This was done specifically to limit the dual-wielding rogue's damage output.)