Which travel documents (passport, ID, etc.) should you carry at all times.
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My general rule is always have ID of some type. Aside from that, I have no issues leaving everything else at the hotel, either in a safe or hidden spot.
If you will be day tripping or taking public transport (other than within a city), then probably best to take your passport. Crossing borders, even in the EU, take your passport.
The Schengen rules abolish the requirement to show your passport when passing the internal borders of Schengen countries. Or even to stop to show anything for that matter when traveling by ground.
Not quite. Nations may do anything they want at their borders. Between France/Germany (west of Saarbrucken), France/Spain (south of Le Boulou), and Spain/Portugal (everwhat the road is that runs into the Algarve), I've been stopped more than once by national police roadblocks in the old border stations -- not just me, everybody. It looked more like a scan for persons or things, but it required a passport flash every time.