If, like Edward Snowden, your passport was revoked or cancelled, how could you travel?

Cuban refugees in the 1960s who were given US residency did not have access to US passports until they became citizens. They were unable to obtain Cuban passports. They were instead issued a passport-like re-entry document by the United States which they used for travel to third countries. If Ecuador or another country issued such a document Snowden could travel on that. Ultimately it is up to the receiving nation to decide whether to admit the traveler, passport or not. Currently, if a refugee reaches U.S. soil he can ask for asylum even though his method of entry was irregular.

After WWII, many refugees were issued non-passport travel documents by the Vatican. There are other examples. There is a "World Citizen" organization which claims their passport is recognized by many countries.

Before 9/11 it was easy to enter Canada with a driver's license and no other proof of citizenship. At the California/Tijuana border, it was similarly the case that if you did not "look Mexican" you were waved through, especially if you were returning on a Friday night after a bout of drinking.

So this is far from unprecedented.

When a travelling companion of mine had their passport stolen in South Africa, the NZ High Commission issued a ETD (Emergency Travel Document). This was a substitute for a passport.

However, if the government has revoked a passport, they're unlikely to issue such a document.

So then you're left with a few other methods.

In some places, national ID cards are fine to travel internationally - including the Economic Community of West African States, most of the EU (and some nearby countries) with the Schengen Area, the Cooperation Council for the Arab States of the Gulf, and Mercosur (in South America).

US Citizens can travel to Puerto Rico with just a photo ID.

According to Wikileaks, Edward Snowden has been issued with a type of emergency refugee document by Ecuador. Several countries issue similar documents, including the UK and even the US, who has a document called an I-94, which many other countries will recognise as a valid form of ID for transiting to the US. Even the International Red Cross is capable of issuing similar documents, although it's up to each country to decide whether to accept these.

Finally, if you're the head of state, like Queen Elizabeth, you may not even need a passport.

All countries have the sovereign right to determine who may enter their territory and what kind of documentation they will require. As an example, EU citizens do not require passports to travel within the EU, even though they are crossing international boundaries.

Ultimately, it's up to the countries of origin and destination of travel to decide what is acceptable documentation. In addition, any country may choose to provide travel documentation to someone, as can some non-government organisations such as the UN.

He was (or still is) in the international transfer part of the airport, which means he didn't go through Russian customs, nor would he have to go through Russian customs to board the plane. Thus it isn't relevant that his passport is revoked. At the gate it's only used as a photo ID, to verify if the person boarding plane is same as the name on the ticket. AFAIK, at that point it's not cross referenced with overseas databases of revoked passports.

Of course he'll have to pass the customs at his destination, but he's going to ask for political asylum, so again it doesn't matter much that his passport has been revoked.

Interestingly, the main purposes of passports is to control who leaves a country and to provide some assurance that you can come back. So the reason countries demand foreigners hold valid passports is to make sure they will be able to leave after their visit and could be easily deported should the need arise. I believe that's also the reason for rules preventing entry to many countries with a soon-to-expire passport.

Without passport, local authorities would typically first need to ascertain their nationality and have the relevant consulate confirm it and issue a laissez-passer. That's why desperate people sometimes try to get rid of their passport or hide it in the hope of defeating immediate deportation (and occasionally also to claim another nationality to get a better shot at asylum).

Therefore, historically, the main reason countries invalidate passports is to prevent people from coming back. For example, during a short time on the night of November 9, 1989, border guards were instructed to invalidate the passport of anybody who wanted to leave the GDR with a special stamp (they quickly gave up and just let people through). In less dramatic situations, people also often have to surrender their passport to make it more difficult to leave, e.g. following a court order.

None of this seems very relevant to Mr. Snowden's situation as he probably does not want to come back to the US for the time being. Invalidating his US passport first and foremost prevents him from doing just that.

However, as long as he holds onto it, it still would look very “valid” to any third country, no matter what US law has to say about it, as there is no legal framework to prevent people from traveling between third countries, no global database of invalid passports and no other mechanism to enforce this supposed “invalidity” (there is an Interpol database of stolen and lost travel documents but it's far from universal).

Once you are stuck somewhere with an expired passport and no hope of getting help from your own country's consular network, your main recourse is to seek another nationality. Failing that, you could try to obtain some sort of travel document from a third country. Several have been created for refugees or stateless people who cannot get a passport from their home countries (Nansen passport, 1951 Convention travel document, 1954 Convention travel document…). They would typically be issued by the country where a person usually resides but, unlike a passport, do not imply that the holder is a citizen. Alternatively, if a country is prepared to grant you entry but not to give you another status right away, you could get a laissez-passer (from the destination country) valid for this one trip.

Of course, none of this really matters for Mr. Snowden. At this stage, it's a safe bet that any decision regarding his fate would be taken at the ministerial level, on a purely political basis.

Your ability to travel between countries (and sometimes within countries) depends on the willingness of the countries in question to allow you to enter and/or leave them.
A passport is one document often required to do one and/or the other, but there's AFAIK no globally active treaty (iow one that every country in existence has ratified) to require one.
Thus, any country may at its discretion allow or deny entry to anyone, with or without a passport from any country (including its own...).
So yes, Snowden and indeed anyone else can legally travel between a pair of countries without a valid passport, IF the countries in question don't demand he has a valid passport to do so.
International refugees can exist because of this. If someone on the run from his government could not gain entry to another country simply because his own country has refused or invalidated his passport, it'd be impossible for refugees to ever flee their governments.

This is possible in most of Europe. The earliest example I know of, is the Nordic Passport Union from 1952, which allowed citizens of the Nordic countries to travel, work and immigrate freely in these countries, without carrying a passport or requiring a VISA.

Since 1995, this has been expanded further, with the Schengen agreement, which removes the borders within the area, allowing citizens of to cross the internal borders without going through a passport control (however, you are required to carry some sort of proof of citizenship, as there are periodic checks).

There are still passport checks in airports, however, as they often simply have an "international" terminal. They have a few gates you can walk through, some of which say "Citizens of EEA countries", "EU and Schengen passports" or something similar, which (if it's even manned) will more or less just glance at your passport and wave you through.

If you still have your physical passport, it is likely they will let you through anyway, unless they actually run a check on it and the country share information with the country of issue. Most often they are simply checking if the passport is genuine and valid, what nationality you have, and whether or not you have a visa, if needed.

It is also possible for a traveller to fly all around the world without a physical passport, as long as he/she doesn't leave the international terminal at the airport and thus need to pass through a passport control.

EDIT:
Research tells me, that since 1923, a similar concept has been in place in the British Isles, called Common Travel Area.

If you have a second passport from another country it is easy and legal to travel on that. Another good reason to get a second passport, if you can.

Other reasons are getting visa-free travel on one passport when your first one doesn't allow it or avoiding "reciprocity fees" that some countries (such as Brazil) charge US and other passport holders but do not charge EU passport holders.

In order not to be responsible the return flight, some airlines on some flights check the passengers’ passports and travel visa routinely. I believe most transatlantic flights to the US check passports at least.

One would expect a certain, perhaps considerable delay, in finding out that a passport had been revoked, especially if the holder was you and me rather than Snowden.