We investigate conditions for the unique recoverability of sparse integer-valued signals from a small number of linear measurements. Both the objective of minimizing the number of nonzero components, the so-called $\ell_0$-norm, as well as its popular substitute, the $\ell_1$-norm, are covered. Furthermore, integrality constraints and possible bounds on the variables are investigated. Our results show that the additional prior knowledge of signal integrality allows for recovering more signals than what can be guaranteed by the established recovery conditions from (continuous) compressed sensing. Moreover, even though the considered problems are \NP-hard in general (even with an $\ell_1$-objective), we investigate testing the $\ell_0$-recovery conditions via some numerical experiments. It turns out that the corresponding problems are quite hard to solve in practice using black-box software. However, medium-sized instances of $\ell_0$- and $\ell_1$-minimization with binary variables can be solved exactly within reasonable time.