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Alejandro Aceves earned his M.S. at the California Institute of Technology in 1983 and his Ph.D. in applied mathematics from the University of Arizona in 1988. Between 1989 and 2008, he moved through the ranks from assistant to full professor of mathematics at the University of New Mexico. In 2008, he joined SMU as a professor and was department chair of Mathematics (2016–2021). He has had visiting positions, most recently Universita Sapienza Italy; University of Limoges, France; University of Erlangen and the Max Planck Institute in Germany; and Columbia University. He has been a visiting scientist at the Los Alamos National Laboratory and an Air Force Laboratory faculty fellow.

His main research area has been in modeling in nonlinear optics and photonics, and he has worked on climate change research. His research has been funded by the U.S. National Science Foundation, Department of Energy and Department of Defense. He has mentored numerous Ph.D. students and postdoctoral fellows. In 2016, he was elected fellow of the Optical Society of America (OSA), and in 2021, he was elected fellow of the Society for Industrial and Applied Mathematics (SIAM). He has served SIAM in many forms and is the current SIAM vice president for science policy.