In combinatorial group testing, the primary objective is to fully identify the set of at most d defective items from a pool of n items using as few tests as possible. The celebrated result for the combinatorial group testing problem is that the number of tests, denoted by $t$ , can be made logarithmic in n when ${d} = {O}(\text {poly}(\log {n}))$ . However, state-of-the-art group testing codes require the items to be tested ${w} = \Omega \left ({\frac {{d} \log {n}}{\log {d} + \log \log {n}} }\right)$ times and tests to include $\rho = \Omega \left ({\frac {{n}}{{d} \log _{{d}} {n}}}\right)$ items. In many emerging applications, items can only participate in a limited number of tests and tests are constrained to include a limited number of items. In this paper, we study the “sparse” regime for the group testing problem where we restrict the number of tests each item can participate in by ${w}_{\max }$ or the number of items each test can include by $\rho _{\max }$ in both noiseless and noisy settings. These constraints lead to a largely unexplored regime where t is a fractional power of n , rather than logarithmic in n as in the classical setting. Our results characterize the number of tests t needed in this regime as a function of ${w}_{\max }$ or $\rho _{\max }$ and show, for example, that t decreases drastically when ${w}_{\max }$ is increased beyond a bare minimum. In particular, in the noiseless case it can be shown that if ${w}_{\max } \leq {d}$ , then we must have ${t}={n}$ , i.e., testing every item individually is optimal. We show that if ${w}_{\max }={d}+1$ , the number of tests decreases suddenly from ${t}={n}$ to ${t} = \Theta ({d} \sqrt {{n}})$ . The order-optimal construction is obtained via a modification of the classical Kautz-Singleton construction, which is known to be suboptimal for the classical group testing problem. For the more general case, when ${w}_{\max }={{ ld}}+1$ for integer ${l}>1$ , the modified Kautz-Singleton construction requires ${t} = \Theta \left ({{d} {n}^{\frac {1}{{l}+1}}}\right)$ tests, which we prove to be near order-optimal. We also show that our constructions have a favorable encoding and decoding complexity, i.e. they can be decoded in $({poly}({d}) + {O}({t}))$ -time and each entry in any codeword can be computed in ${poly}(\log {n})$ memory space. We finally discuss an application of our results to the construction of energy-limited random access schemes for Internet of Things networks, which provided the initial motivation for our work.