Lost US citizenship certificate, have no U.S. passport - can I travel internationally?

When I posted my comment, I did not have time for a more comprehensive answer. Here it is.

What can I do?

You can apply for a passport. Depending on the date of your intended travel, you may want to apply for expedited service. The US State department has a web page explaining this option in fairly clear terms: http://travel.state.gov/content/passports/english/passports/services/expedited.html

How can I prove that I am a [US] citizen and leave and come back to the US.

Except for certain travel covered under the Western Hemisphere Travel Initiative (WHTI), US citizens are required by law to use a US passport when entering and leaving the country. There is apparently no penalty for violating the law, but you will likely encounter an administrative headache if you try to enter with just a citizenship certificate. There is also a risk that the border officer might be unable to establish that your certificate is genuine, in which case you might be refused entry (the certificates are presumably more easily forged than passports).

More pressingly, however, if you are attempting to travel by air, you will most likely be unable to board the airplane, and you will be stuck in that country until you get a new passport from the US consulate. The only exception I can think of would be if you are also a Canadian citizen, since Canadians do not need a visa to enter the US.

Another possibility would be if you are a citizen of a visa waiver country, in which case you could get ESTA authorization to travel with your other passport. But you're not supposed to do that if you're a US citizen (plus, if you were planning to do that, you probably wouldn't be worried about your citizenship certificate).

The WHTI applies only to land and sea travel. If your travel is by land or sea, you can travel without a passport, but there are other identification requirements for which a citizenship certificate does not suffice. The documents you can use are typically secure identification cards, including the US "passport card," so-called "enhanced" driver's licenses, and secure traveler program cards such as NEXUS. The full list can be found here: http://www.cbp.gov/travel/us-citizens/whti-program-background/docs-land-sea.

I thought that there was a list of countries to which the WHTI applied, and that one would require a passport if one were arriving by sea from, for example, Europe or Asia, but I don't see that information on the CBP site: http://www.cbp.gov/travel/us-citizens/whti-program-background.

To get your passport, you will probably need to present your citizenship certificate, though a naturalization certificate will do as well, so you should probably work on getting it replaced as soon as possible.

Ignoring all the other "exceptions", the bottom line is:

To travel internationally, you need a passport.

Now the situation comes how do you prove that you are citizen of the US since you have misplaced your citizenship certificate. You need to fill out form N-565 and get a replacement certificate. It costs $345.

Take the replacement certificate and apply for a passport.