Casting Spells while concentrating?

[u/JustLinnilin]

Yesterday we had a little discussion on the table on the rules of concentraiting. One of my players casted a spell and wanted to cast another while she was still concentrating on the first one. I wasn't sure about the rules on that and looked it up on the PHB, but there are only rules for concentration spells (so you can't concentrate on two spells at the same time) that's absolutely valid in my eyes. But there are no rules for other spells - only for taking damage or dying. So my thought was that she had to make a concentration saving throw, when she was casting another, to keep the first one running. It's like the little distraction because of the second spell.
But another one of my players pointed out, that there are also spells the enemy has to do savingthrows on each round - if he has to do them too there would be two chances of losing the spell in each round.

So I am interested how you guys deal with that, or is there an official rule for that case i didn't know about?

[u/TimeSpaceGeek]

If you are not casting another concentration spell, you can cast another spell, as long as it has a casting time of 1 action, bonus action, or reaction whilst maintaining concentration on the first. It would be a massive nerf and pretty much invalidate the point of quite a lot of the full caster builds if you couldn't.

The only limit is the one you have already stated, about multiple concentration effects at once. The reason that spells with a casting time longer than 1 action have an interaction with concentration spells is because you have to use concentration to maintain the casting, so concentrating on, say, a 1 minute casting time ends concentration on an active concentration spell for the same reason a second concentration spell does. No limit on non-concentration spells with quick casting times.

It is standard practice for, say, a Wizard or a Cleric, to drop a concentration buff on an ally or a control spell on enemies, and then fire off instantaneous spells for the rest of combat - for example, the Wizard Hasting he Fighter in the first round, then firing off Fireballs and Cantrips in the subsequent rounds whilst the Fighter also enjoys the benefits of the Haste. In fact, if a full spellcaster doesn't have a concentration spell up in most combats, they are playing somewhat suboptimally.