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When do I need to carry my passport. When I am within walking distance of where I am staying, to another close city on a day-trip, etc.

Italy

Italy's Passport Policies. Italy requires you to carry official I.D. ... This means that, in Italy, even though you're unlikely to be checked, you must have your passport with you at all times.

You should carry it everywhere you go:

Can be checked at random traffic stops, at which the drivers of cars, scooters etc are obliged to produce the appropriate documentation, though I've never noticed the passengers being asked to show theirs!

Switzerland

Foreign nationals must be in possession of a valid identity document recognised in terms of Article 13 paragraph 1 during their stay in Switzerland. Conclussion: You have to posses but not to carry a valid form of ID.

You must possess it but not carry it everywhere.

This is much more the case now that there are no border controls between the surrounding EU countries and Switzerland... as a result there are sometimes (note: very infrequent!) border control checks WITHIN Switzerland... and if they ask you for ID you must be able to produce it.

France

Before you leave home, make two or three copies of the pertinent pages of your passport. ... Contrary to what many believe, you are not required to carry your passport with you—a photocopy will suffice.

You don’t need to carry it and the police will give you two days to supply it:

No need to carry your passport. I have lived in Paris for 40 years and never been asked for identity papers in the street. If you are asked and don't have them, you have 48 hours to supply them at the police office.

Netherlands

The law requiring you to carry a passport on you at all times is only enforced by the police. Coffeeshops are not required by law to ask for a passport, only proper age ID, but some insist on passports. You only have the police to fear for a fine if you don't have your passport on you.

You don’t need to have it anywhere:

As long as you have an absolutely secure place to keep it (hotel safe), you will not need it on you at all times.

UK

Should I carry a passport around everywhere I go in the UK?

The UK is not like Russia or Uzbekistan where police on the street can and do stop you without cause and demand ID (in my case, six times in one day in Tashkent, Uzbekistan). And in the event they DO need your ID, a driver's license generally suffices - or they can go with you to your place of residence to get your passport if you for some reason got into serious strife.
I can only think of a handful of cases that I needed my passport. To open a bank account, to prove my visa status for starting a new job, for renting a flat / staying in a hostel, and you definitely need it for hiring a car. I remember this one as I tried to argue it, but had to eventually go back home and get my passport. I wasn't pleased.

You don’t need to carry it but it is recommended.

In general you should carry it if you are not sure just in case.

I am a black man of African descent who visits Europe extensively (Italy, Germany, Spain, virtually all the Schengen countries + UK).

I am a prime candidate to be stopped by the authorities to check my papers because of the influx of undocumented African migrants. However it has never happened.

I never carry my passport on my person in Europe, it’s always safely secured at my hotel.

I mean what’s the worst that will happen to you in the improbable scenario when you’re asked by the authorities in town?

With every rule there’s theory, and then there’s practice. This is one rule where I believe practice (not carrying your passport around) is the norm.