Graduate admissions play a critical gatekeeping role in the physics community not only because they select students who are allowed to begin their graduate studies, but also because they influence how students perceive graduate school, and in some cases whether or not they will even choose to apply. In conjunction with the APS Bridge Program, we conducted a national survey of graduate directors (and related faculty) of physics Ph.D. programs in the United States to explore graduate admissions practices. Our focus was on criteria used in determining admissions, mechanisms through which graduate applicants are handled, and how student representation considerations are incorporated into admissions (if at all). We report here on existing graduate admission practices in physics departments and highlight some critical issues for understanding barriers for diversifying graduate physics, including the use of GRE scores (and the relative importance placed on them). We find that the use of a minimum GRE score for admission, a practice in opposition to recommendations made by the tests designers, is reported to be used in many departments (more than one in three). We also find letters of recommendation to be highly valued in admissions decisions. Our data describe various initiatives at the institutional or individual level to increase gender diversity in admissions. A sizable number of departments also express a latent demand for greater numbers of students from traditionally marginalized racial or ethnic groups, but simultaneously report a lack of such applicants.