The American Astronomical Society (AAS, see here for more information) keeps a list of universities that have astronomy programs. You can find that list here with links to the various department webpages. Most of the schools on this list are US institutions, but a few are international. When choosing the graduate schools that you will apply to, you might want to keep a few things in mind:    

• Does the department do research that interests you? The research areas that a department focuses on can usually be found on the department website. 
• The school might have the general topic you are interested in, but are there faculty members doing specific research that interests you? 
• Is the school in a place you could see yourself living for the next 5 years?
• Is the department a stand-alone astronomy department, or is it combined with a physics department? 
• How big is the department? How many grad students, post-docs, and faculty members?
• Do you like the format of the graduate program? For example, some departments will only require that students take astronomy classes, others will require you to take classes in physics. Be sure to find out what you will be tested on in your qualifying exams. This is particularly relevant if it is a stand-alone astronomy department, or a joint physics and astronomy department. 

When you start looking for grad schools, make your own list of things that are important to you. The criteria you define will help you to narrow down the list of schools to apply to. 

 

If you are interested in attending graduate school in another country, look here for advice from Yvette Cendes, a graduate student in Amsterdam who is originally from the US.