This paper presents the topological invariants of time-reversal-invariant (T-invariant) band structures in two and three dimensions. In two dimensions, these invariants are multiple copies of the Z₂ invariant introduced by Kane and Mele, which protects the topological insulator and gives rise to a spin Hall effect via edge states. Each pair of bands related by time reversal is described by a single Z₂ invariant. In three dimensions, there are four such invariants per band. These invariants determine the transitions between ordinary and topological insulators as bands are filled with electrons. The Z₂ invariants are derived using maps from the Brillouin zone to the space of Bloch Hamiltonians, clarifying their connection to the integer invariants underlying the integer quantum Hall effect (IQHE) and previous invariants of T-invariant Fermi systems. The Z₂ invariant is rederived for the simplest case of two occupied bands related by time-reversal symmetry using homotopy theory, establishing its connection to the usual Chern number or TKNN integer. For 2D T-breaking systems, each band is associated with a TKNN integer that is invariant under smooth perturbations of the Bloch Hamiltonians. The TKNN integers can be obtained as integrals of Berry flux or using projection operators. The existence of these integers follows from the first two homotopy groups of the space of Hamiltonians. The paper also discusses the implications of time-reversal symmetry on the topology of the Brillouin zone and the resulting invariants. It shows that for a degenerate band with time-reversal symmetry, the Chern number for the whole Brillouin zone vanishes, but there is still a topological invariant. The paper generalizes these results to three dimensions, where there are four independent Z₂ invariants per pair of bands, even though there are only three Chern numbers for a pair of degenerate bands. The paper concludes by discussing the implications of these results for the understanding of topological insulators and their connection to the quantum Hall effect. It also notes that the results can be applied to many-body problems with an odd number of fermions if there are two periodic parameters in the Hamiltonian connected by time-reversal symmetry. The paper highlights the robustness of the topological insulator phase and its potential applications in a wider class of materials, especially in three dimensions.This paper presents the topological invariants of time-reversal-invariant (T-invariant) band structures in two and three dimensions. In two dimensions, these invariants are multiple copies of the Z₂ invariant introduced by Kane and Mele, which protects the topological insulator and gives rise to a spin Hall effect via edge states. Each pair of bands related by time reversal is described by a single Z₂ invariant. In three dimensions, there are four such invariants per band. These invariants determine the transitions between ordinary and topological insulators as bands are filled with electrons. The Z₂ invariants are derived using maps from the Brillouin zone to the space of Bloch Hamiltonians, clarifying their connection to the integer invariants underlying the integer quantum Hall effect (IQHE) and previous invariants of T-invariant Fermi systems. The Z₂ invariant is rederived for the simplest case of two occupied bands related by time-reversal symmetry using homotopy theory, establishing its connection to the usual Chern number or TKNN integer. For 2D T-breaking systems, each band is associated with a TKNN integer that is invariant under smooth perturbations of the Bloch Hamiltonians. The TKNN integers can be obtained as integrals of Berry flux or using projection operators. The existence of these integers follows from the first two homotopy groups of the space of Hamiltonians. The paper also discusses the implications of time-reversal symmetry on the topology of the Brillouin zone and the resulting invariants. It shows that for a degenerate band with time-reversal symmetry, the Chern number for the whole Brillouin zone vanishes, but there is still a topological invariant. The paper generalizes these results to three dimensions, where there are four independent Z₂ invariants per pair of bands, even though there are only three Chern numbers for a pair of degenerate bands. The paper concludes by discussing the implications of these results for the understanding of topological insulators and their connection to the quantum Hall effect. It also notes that the results can be applied to many-body problems with an odd number of fermions if there are two periodic parameters in the Hamiltonian connected by time-reversal symmetry. The paper highlights the robustness of the topological insulator phase and its potential applications in a wider class of materials, especially in three dimensions.