Does D&D 5e lack a general disarm rule?
I get from the PHB that there is no general rule for disarming an opponent but only a specific rule for Battle Master fighters.
Can we assume that other classes (even other fighters subclasses) lack the training for such move or can we deduce that everyone can attempt a disarm without the superiority dice?
The closest thing to a general rule I can see is the sidebar in the section on Grappling/Shoving a Creature (Basic Rules p.74):
Contests in Combat
Battle often involves pitting your prowess against that of your foe. Such a challenge is represented by a contest. This section includes the most common contests that require an action in combat: grappling and shoving a creature. The DM can use these contests as models for improvising others.
It certainly seems like the intention of a section like this would be to allow you to attempt special maneuvers (such as disarming) as an action in combat via a contest if you're not as trained as the Battlemaster, but, as always, it's up to the DM.
As of the DMG's release (DMG p.271), there is an officially suggested contest for disarm attempts (though, to be clear, it is still an optional rule/action that may or may not be present in any given DM's game):
A creature can use a weapon attack to knock a weapon or another item from a target's grasp. The attacker makes an attack roll contested by the target's Strength (Athletics) check or Dexterity (Acrobatics) check. If the attacker wins the contest, the attack causes no damage or other ill effect, but the defender drops the item.
The attacker has disadvantage on its attack roll if the target is holding the item with two or more hands. The target has advantage on its ability check if it is larger than the attacking creature, or disadvantage if it is smaller.
I think this is a good way to handle it. Moreover is less effective than a Battlemaster manouver (@Joshua) which inflicts considerable damage and has a harder saving throw.
+1 for sufficient ambiguity. I think this side-bar opens the doors for Players and DMs alike to new and exciting combat capabilities; ones we've all seen and used before if we've ever played 3.5e. This basically spells out the possibilities of disarming and tripping and possibly feinting and sundering as well. It's a gosh darn shame the spiked chain is gone (spiked chain fighters were ridiculous anyway).
So essentially the weapon is dropped on the ground? Will it take their next action to pick it up?
@BrightIntelDusk I believe the rules would treat retrieving the weapon from the ground as the 'free object interaction' for the turn (for the disarmer or whoever else wished to retrieve it). If they didn't wish to use their one free object interaction on retrieving it, then it would take an action. That would be better off as (or may already be) its own question if the details are a concern, however.
If the weapon is on the ground then an adventurer in the same party could potentially pick up the disarmed weapon, use their action to disengage and run off with the weapon. Especially if their turn in combat is before the enemy gets a turn. This would leave the enemy to pick a different weapon or modify their attack to be a hand to hand combat. Is this correct?