My husband left the country of the US and doesn’t pay alimony. What do I do?

There may be very little you can do as a practical matter. Most European nations will honor a U.S. judgment or court order, BUT the process is expensive and cumbersome: you would have to file a new lawsuit in Europe typically (hiring a European attorney) to do so, which legal action is based on violation of a U.S. order order or decree, then after you (presumably) win that case, use whatever collections methods that country provides for (since you are asking its courts to enforce your judgment or decree, you are limited to the methods, powers, or techniques that country itself provides for collections: they cannot and will not do anything not permitted under their law). So you're talking about taking legal action overseas, with an overseas attorney.
The above is a huge oversimplification: the process, the costs, how "cooperative" the courts will be, etc. vary nation by nation. Some are easier: the UK, not suprisingly, is one of the easier ones, because their law and ours are very similar (we got our oldest source of law, the "common law," from the UK back during Colonial days) and because of the nations' close relationship. Other countries, such as the former Soviet republics, are apt to be much more difficult, due to very different legal traditions and different attitudes towards the role of the courts. Even France or Germany, which are close allies of ours, have a different legal tradition and structure ("civil" law, not common law) than we do; and some other friendly countries (e.g. Greece) may be in such disarray right now as to make working with their courts difficult or time consumming.
You also need to know where he is (e.g. his address) or be able to find it.
If you want to pursue the matter, you will have to either contact a large U.S. law firm with offices in the country where you ex is, or similarly, a big international firm with offices in the U.S. and that other countr (either of which is probably the best option, since you will be able to deal with the U.S. office, improving communication and coordination), or else find an attorney directly in that other nation to help you.