I would like to connect two Arduinos by a serial connection. Is it sufficient to connect TX1<-->RX2 and RX1<-->TX2 or do I also need to connect the GND of the two Arduinos?
Short answer: Yes.
Rx/Tx is referenced with a logic ground. Without ground being common, there is no way to determine what is high or low.
(Sorry I don't have a more technical explanation)
Edit: I thought I would add, if you are using the same power source, then they probably have a common ground. In this case, a common power source doesn't mean the wall side of a wall adapter, but the SAME wall adapter.