This past fall/winter I applied to 16 of the pure math summer REU programs listed by the AMS here, and haven't been accepted by any so far. Five have rejected me, and the fact that none of the other 11 have sent me information tells me I'm probably on their waitlists.
I'm in my junior year and was hoping to apply to a PhD program next fall. I have a very high GPA, but I haven't taken a really competitive course load since I have been struggling with anxiety/depression for several years. It wasn't until my junior year began that I started the year-long abstract algebra and real analysis sequences. I'm planning to take grad math classes in my senior year, and ask some professors if I can get involved in research (haven't done any thus far).
I'm starting to worry about what I'm going to do this summer if I get rejected from all of the REU's... any advice? And, does this mean that I should look elsewhere instead of pursuing graduate school in pure math? I guess I'm feeling a bit discouraged about the prospects of a career in pure mathematics.
Mid-summer update: I actually did end up getting into an REU this summer, and I'm loving it here! At the same time, I'm really glad I asked this question, since it was important (both emotionally and logistically) to be prepared for the possibility of not getting into an REU. Interestingly, the professor who I'm working with told me that he never went to an REU as an undergrad; he didn't start doing research until his 3rd year of grad school. Somehow I had convinced myself that only students who do undergraduate research will become successful mathematicians. Now I understand that while doing undergraduate research can help students to delve deeper into specific topics, beef up their resumes, and prepare for graduate school, it's definitely not the "be-all and end-all" of a mathematician's career. Thank you for the advice, and good luck to all of the other students out there!