One of the most disappointing aspects of Diablo 3 is its bare-bones approach to player-versus-player. Unfortunately, that side of the game might not improve anytime in the future.
A Polish Diablo 3 player asked on the Battle.net forums asked about upcoming features for the game. Here's the response from a community manager:
I remind you that if you want to fight with other players, then of course you can do it on a separate map for some time, we currently do not plan further development of skirmish mode.
As DiabloFans points out, this statement hasn't been confirmed by the North American community team. However, without any other statement from Blizzard contradicting this one, we have to take this at face value.
Diablo 3 didn't launch with any player-versus-player activities. Months afterward, they scrapped a Team Deathmatch mode because combat with no objectives "gets old quickly." Their alternate plan was to create competitive modes with firm goals.
"We're going to first be looking at new modes that play up to the strengths of the character classes, focus on objectives beyond just defeating other players, and possibly even integrate PvE elements and rewards," then-director Jay Wilson said.
That sounded like a great plan. However, what Blizzard ended up producing instead of Team Deathmatch was a mode called Brawling...which was pretty much just Deathmatch. Brawling supports one-on-one duels as well as three- or four-player free-for-alls. There's no scoring or ranking system. At best, it's a momentary diversion.
I had really hoped that Blizzard would return to the objective-based PvP idea. Even something as unimaginative as a 4v4 capture-and-defendor capture-the-flag mode would be a dramatic improvement over the current state of player-versus-player. Even a ladder for Brawls would've been appreciated.
However, it seems that Blizzard has decided that they're only going to bother with PvE endgame. If you want to keep advancing your character after beating the campaign, you're supposed to play Adventure Mode. The only organized competitions in the game are the Seasons. You're not battling other players in Seasons, though; instead, you're merely trying to outperform others in the campaign.
If you're looking for more extensive player-versus-player, you should check out Path of Exile. PoE holds regular PvP events that award in-game items to the top-performing players. Certain Leagues in the game also allow players to hunt each other down while they're questing. On top of all that, PoE is free to download and play.