Astrobiology in SESE
About 50 SESE faculty and students are members of ASU's team in the NASA Astrobiology Institute ("NAI"); more information on this group's research themes can be found at http://astrobiology.asu.edu. Your own research need not lie along the same lines, but you will very likely find yourself working with some of the NAI center members.
Your Background
This page by our NAI group's P.I., Ariel Anbar, states that: "We ... encourage applications from students with B.S. or M.S. degrees in geology, biology, physics, chemistry, engineering fields and related areas."
Whatever your background — be it astrophysics or geology or something else — you will feel at times in-over-your-head in one discussion or another, especially if it is outside your field. Remember, the faculty here and your fellow students are enthusiastic about their work and will be happy to explain it further.
Sometimes it may seem the astrophysicists keep to themselves, and the geologists keep to their side, too; this may have been true at your undergraduate institution, but it is becoming less the case in SESE with the growing involvement in the NAI center and other interdisciplinary groups.
Many faculty and students blur the former distinctions between fields, and even when their own research is firmly planted in one, their skill-set may well be applicable to others. If your background is something outside SESE's nominal coverage, you may find yourself sought out for the external knowledge and skills you bring to the department.
Requirements
Officially you will be enrolled in either an Astrophysics or Geological Sciences degree program, and not as an "Astrobiology MS/PhD" student (at present, at least). In deciding your specific research topic(s) you may find yourself involved more heavily in one — astrophysics or geology — than the other, or equally in both. You may find you need to officially change degree programs if your research lies mostly in one field which is not the one you originally entered; mechanically, this can amount to just filing a few forms.
When you have chosen a degree program that best suits your research and goals, you will need to satisfy that program's requirements (e.g., X hours' coursework, qualifying oral examination, technical review, submission of a thesis/dissertation, etc.); however, since your work may well blur the lines between astrophysics and geology, you may petition to have certain courses count towards fulfillment of one degree or another. For example, the Astrophysics programs prescribe a specific set of courses to take, AST 521/522/523 and 531/532/533; you can argue that your own work is better served substituting, say, biogeochemistry or planetary geology for some of those AST courses. This "petition" may be just a matter of discussion with your advisors, or may take the form of a formal written document (the latter may be necessary if you end up formally switching from one degree program to another).
Note that graduate program policies and procedures may change with time, so keep in contact with your advisors and the graduate committee to stay up-to-date.
Main point of this: your program of study is yours to make. 
Discuss options with your advisors and the graduate program committee and work out a plan that both satisfies the program requirements and your own goals and interests in astrobiology.
Courses
Given the multi-disciplinary nature of astrobiology, you may find courses offered in SESE and in other departments that suit your interests. Below are a few courses taken by astrobiology students at ASU in the past. These are not offered every semester, or even every year, so you will want to find out in advance what is expected to be open for a given term:
Courses through SESE:
(some may be cross-listed between AST/GLG or in other departments like CHM and SOS)
GLG 485/598: Meteorites and Cosmochemistry (3 cr)
Covers a lot of topics of interest to planetary system evolution, meteorites, and evolution of the chemical elements — a good one for folks associated with ASU's NAI team.
GLG(?) 598: Astrobiology Seminar (3 cr)
In spring 2008 there were only three of us in the class and it became somewhat of an informal, make-your-own-course — in addition to some unifying readings and discussions, each of us chose a topic to delve into and present to the others. In other semesters it may be more structured. In our case, we were three astronomy students with the instructor, a geologist/astrobiologist, and we spent a good portion of the course filling in some missing geology and planetary sciences background.
AST 522: Stars and the Interstellar Medium II (3 cr)
In fall 2008 it centered on stellar evolution and nucleosynthesis which was great for the author of this page, given my research. Like Meteorites and Cosmochemistry, it could be very useful for students affiliated with the NAI group, but may seem of less interest/importance to others (non-astrophysics programs).
GLG 430/598: Paleontology (3 cr)
Covers the fossil record, important factors in preservation, and evolution in Earth's biology, of course, but also delves well into earlier terrestrial climates and habitats, and life in extreme environments.
GLG 404/598: Planetary Geology (3 cr)
Overview of planetary geology, with guest lectures by other faculty on astrobiology and other topics. Also included, at least in fall 2007, was a wealth of information on the business of planning and operating planetary exploration and other missions.
GLG 490: Science for Sustainability (3 cr)
Note the "for" instead of "of" in the course title. This was not a course on economics or social sciences, policy, etc.; instead, the course (as it was taught in spring 2008) focused on biogeochemical cycles and climate change, with discussion of human impacts and ideas for management thereof. This course's interest to astrobiologists may be in the biogeochemistry and questions about the lifetime of habitable environments.
Courses in other departments: (thanks to Jess Corman in the School of Life Sciences for these!)
MIC 461: Geomicrobiology
Past and present interactions among microbial life, geological materials, and biogeochemical cycles involving carbon, sulfur, phosphate, nitrogen, and metals.
MIC 470: Bacterial Diversity and Systematics
Biology, classification, and enrichment culture of the nonpathogenic bacteria.
BIO 524: Ecosystems (Every other fall, 3 cr)
Structure and function of terrestrial and aquatic ecosystems, with emphasis on productivity, energetics, biogeochemical cycling, and systems integration.
BIO 460/MIC 475: Astrobiology
Origin, early evolution, distribution, and future of life on Earth and elsewhere in the cosmos.
BIO 543: Molecular Genetics
Nature and function of the gene; emphasis on the molecular basis of inheritance and gene expression in procaryotes and eucaryotes.
MIC 598 & BIO 598:
Special topic seminars that may be of interest
CHM/GLG 598: Advanced Biogeochemistry (fall, 3 cr)
Lecture and discussion-based course on biogeochemistry, the interdisciplinary science that integrates chemistry, geology, and biology to study processes that shaped ancient systems and those at work in modern environments.
