Group (pooled) testing, i.e., testing multiple subjects simultaneously with a single test, is essential for classifying a large population of subjects as positive or negative for a binary characteristic (e.g., presence of a disease, genetic disorder, or a product defect). While group testing is used in various contexts (e.g., screening donated blood or for sexually transmitted diseases), a lack of understanding of how an optimal grouping scheme should be designed to maximize classification accuracy under a budget constraint hampers screening efforts. We study Dorfman and Array group testing designs under subject-specific risk characteristics, operational constraints, and imperfect tests, considering classification accuracy-, efficiency-, robustness-, and equity-based objectives, and characterize important structural properties of optimal testing designs. These properties provide us with key insights and allow us to model the testing design problems as network flow problems, develop efficient algorithms, and derive insights on equity and robustness versus accuracy trade-off. One of our models reduces to a constrained shortest path problem, for a special case of which we develop a polynomial-time algorithm. We also show that determining an optimal risk-based Dorfman testing scheme that minimizes the expected number of tests is tractable, resolving an open conjecture. Our case studies, on chlamydia screening and screening of donated blood, demonstrate the value of optimal risk-based testing designs, which are shown to be less expensive, more accurate, more equitable, and more robust than current screening practices. / PHD / Group (pooled) testing, i.e., testing multiple subjects simultaneously with a single test, is essential for classifying a large population of subjects as positive or negative for a binary characteristic (e.g., presence of a disease, genetic disorder, or a product defect). While group testing is used in various contexts (e.g., screening donated blood or for sexually transmitted diseases), a lack of understanding of how an optimal grouping scheme should be designed to maximize classification accuracy under a budget constraint hampers screening efforts. We study Dorfman and Array group testing designs under subject-specific risk characteristics, operational constraints, and imperfect tests, considering classification accuracy-, efficiency-, robustness-, and equity-based objectives, and characterize important structural properties of optimal testing designs. These properties provide us with key insights and allow us to model the testing design problems as network flow problems, develop efficient algorithms, and derive insights on equity and robustness versus accuracy trade-off. We also show that determining an optimal risk-based Dorfman testing scheme that minimizes the expected number of tests is tractable, resolving an open conjecture. Our case studies, on chlamydia screening and screening of donated blood, demonstrate the value of optimal risk-based testing designs, which are shown to be less expensive, more accurate, more equitable, and more robust than current screening practices.