Several other establishments offer a variety of vegan foods. Just peruse the menu, ask questions and let your server know that you are seeking options that contain no meat, poultry, fish, dairy or eggs. In addition to ensuring you get what you want, this will also let restaurants know there is a demand and a market for animal-free fare. Call ahead of time if you don’t wish to conduct an inquiry on-the-spot.

Some general guidelines on what to watch out for and ask about:

Chinese: Watch out for chicken broth (even in “vegetable” dishes), eggs (often in soups, egg roll wrappers, lo mein noodles. Ask for egg left out of veggie fried rice and moo shu).
Ethiopian: Most vegetarian dishes are usually vegan, but dairy is occasionally used (usually in the form of butter or a fresh, soft cheese).
Indian/Nepalese: There are usually many vegetarian options, just beware of dairy which can pop up in many forms, especially paneer (cheese), ghee (clarified butter), yogurt, and cream (in many sauces, breads, chutneys, drinks, and even rice sometimes).
Italian: Most dried pastas are vegan, however restaurants which make their own pastas tend to use egg in the dough. Some tomato sauces may contain animal broth or dairy products (cheese), most marinaras are okay, though. Olive oil and garlic is a good alternative if you find yourself stuck. Likewise a lot of dried gnocci is vegan, but most fresh made gnocci are not.
Japanese: Ask about fish sauce or bonito (fish flakes). These can show up in everything from seaweed salad to miso soup and beyond. You can usually at least get some vegan sushi rolls (maki), and an udon (wheat noodle dish) or two at a sushi restaurant .
Mexican: Watch for chicken stock in rice, lard in beans and tortillas, and dairy (sometimes sour cream is used) in guacamole. Plus ask to leave off the cheese/sour cream where listed, obviously.
Middle Eastern: Usually felafel, hummus, tabouleh, fattoush, muhammara (walnut spread with red pepper and/ or pomegranate) and pita are okay. Baba ganoush and tahini sauces may or may not be dairy-free (some contain yogurt). Tzaziki contains yogurt.
Thai: Look out for egg (especially in pad thai and other noodle dishes and rice dishes) and fish sauce (nam pla) or shrimp paste in curries and other items.
Vietnamese: Watch out for eggs and especially fish sauce which can be used widely and may not be listed on menus. Even dishes listed as “vegetarian”  may typically come with fish products, as the word is understood differently culturally.