From D&D 3.5 to 5.0(5E) - 18 - Spell Concentration

I'm not sure what the rules were in 3.5, but we always played that you could not cast a second spell while you are still concentrating on the first spell you cast, without giving up the first spell.  But in the 5th edition it is clear from page 203 that you can cast additional spells while continuing the concentration of the first spell, as long as the subsequent spells do not also require concentration.

Normal activity such as moving and attacking (I think it is funny that attacking is "normal activity") doesn't break your concentration.  But getting incapacitated does, so does getting killed, and so does casting another spell that requires concentration.  Getting hit in combat may also cause you to lose concentration if the damage is severe enough.