Since 2005, UCLA has run an internal NSF-funded summer REU program in applied mathematics for talented UCLA students and, more recently, students from other local colleges. The REU program has been very successful and is continuing to evolve into a better program. The unique feature of this program is that the undergraduate research projects are intrinscially tied into ongoing research carried out by the faculty and graduate students. I will discuss my involvement with the REU program for the last 3 years and present some of the projects I have mentored.The goal is to suggest a possible template for other schools to develop their own REU program in mathematics.