As many schools and departments are considering the removal of the Graduate Record Examination (GRE) from their graduate admission processes to enhance equity and diversity in higher education, controversies arise. From a psychometric perspective, we see a critical need for clarifying the meanings of measurement bias and fairness, in order to create common ground for constructive discussions within the field of psychology, higher education, and beyond. We critically evaluate six major sources of information that are widely used in graduate admission assessment: grade point average, personal statements, resumes/CVs, letters of recommendation, interviews, and GRE. We review empirical research evidence available to date on the validity, bias, and fairness issues associated with each of these admission measures, and identify potential issues that have been overlooked in the literature. We conclude by suggesting several directions for practical steps to improve the current admissions decisions, as well as highlighting areas in which future research would be beneficial.