While many PhD admissions criteria demand the reporting of GRE scores, it's not always required. However, the student must first ascertain whether her field of interest usually requires either the general test or a more specific subject test. In such cases, reporting is likely required.
There are many PhD fields that typically require the GRE general test, ranging from the humanities and social sciences to business degrees. Additionally, some students who have dedicated themselves to certain fields must take a more specific GRE subject test, such as in Biochemistry, Biology and Physics. Other fields commonly requiring subject tests include mathematics, psychology, chemistry and literature in English.
Thus, PhD programs that do not require GRE reporting are often more exceptions to the rule rather than the opposite. For example, the prestigious Northwestern University provides no GRE requirement for its PhD program in history, an extremely rare exemption in that field. Similarly, Towson, Purdue, New Mexico State and the University of Maryland have no GRE requirement for their programs in computer science and software engineering.
Other schools, such as Columbia, Boston College, Oregon State and the University of Pennsylvania, exempt applicants to the graduate programs in education. Nevertheless, even though a program might not officially require a GRE score for admission, the prospective student should still ascertain whether submitting a score makes them more competitive. Proper dialog with an informed adviser is recommended.