This paper presents a study of the electronic structure of twisted double-layer graphene (t-dLG), focusing on the formation of moiré bands and their dependence on the twist angle. The authors develop a continuum model to describe the system, showing that the moiré pattern leads to the formation of moiré Bloch bands. As the twist angle decreases, the two layers become more coupled, and the Dirac velocity crosses zero several times. At specific "magic angles," the velocity vanishes, the lowest moiré band flattens, and the Dirac-point density-of-states and counterflow conductivity are significantly enhanced. The electronic properties of few-layer graphene (FLG) are strongly influenced by the stacking arrangement. In bulk graphite, specific twist angles lead to rhombohedral or Bernal crystals, but other angles are also observed. Small twist angles are common in epitaxial graphene on SiC, and arbitrary layer alignments can be achieved by folding a single graphene layer. Recent theoretical studies have focused on the electronic properties of FLG with arbitrary twist angles, particularly the two-layer case. The problem is mathematically interesting because a bilayer forms a two-dimensional crystal only at specific commensurate angles. For generic angles, Bloch's theorem does not apply, and direct electronic structure calculations are not feasible. At larger angles, the layers are nearly isolated, except at specific angles that yield low-order commensurate structures. As the twist angle decreases, interlayer coupling increases, and the quasiparticle velocity at the Dirac point begins to decrease. The authors construct a low-energy continuum model Hamiltonian for t-dLG, consisting of three terms: two single-layer Dirac-Hamiltonian terms and a tunneling term. The Dirac Hamiltonian for a rotated layer is given, and the continuum model for the tunneling term is derived by assuming that the inter-layer tunneling amplitude is a smooth function of separation. A π-band tight-binding calculation yields a continuum limit where tunneling is local. The authors derive a continuum model for the tunneling term and find that the amplitude for an electron to hop between sublattices is given by a specific expression. By comparing with the known electronic structure of an AB-stacked bilayer, they set the tunneling amplitude to a specific value. The spectrum is independent of the separation d for θ ≠ 0. The continuum model captures the local stacking sequence of the misaligned layers, and the periodicity is controlled by reciprocal lattice vectors. The authors solve numerically for the moiré bands using the plane-wave expansion and find that the velocity at the Dirac point oscillates with the twist angle, vanishing at a series of magic angles. These angles give rise to large densities-of-states and large counterflow conductivities. The authors also find that the Dirac-point velocity vanishes at θ ≈ 1.05°, and that the vanishing velocity is accompanied byThis paper presents a study of the electronic structure of twisted double-layer graphene (t-dLG), focusing on the formation of moiré bands and their dependence on the twist angle. The authors develop a continuum model to describe the system, showing that the moiré pattern leads to the formation of moiré Bloch bands. As the twist angle decreases, the two layers become more coupled, and the Dirac velocity crosses zero several times. At specific "magic angles," the velocity vanishes, the lowest moiré band flattens, and the Dirac-point density-of-states and counterflow conductivity are significantly enhanced. The electronic properties of few-layer graphene (FLG) are strongly influenced by the stacking arrangement. In bulk graphite, specific twist angles lead to rhombohedral or Bernal crystals, but other angles are also observed. Small twist angles are common in epitaxial graphene on SiC, and arbitrary layer alignments can be achieved by folding a single graphene layer. Recent theoretical studies have focused on the electronic properties of FLG with arbitrary twist angles, particularly the two-layer case. The problem is mathematically interesting because a bilayer forms a two-dimensional crystal only at specific commensurate angles. For generic angles, Bloch's theorem does not apply, and direct electronic structure calculations are not feasible. At larger angles, the layers are nearly isolated, except at specific angles that yield low-order commensurate structures. As the twist angle decreases, interlayer coupling increases, and the quasiparticle velocity at the Dirac point begins to decrease. The authors construct a low-energy continuum model Hamiltonian for t-dLG, consisting of three terms: two single-layer Dirac-Hamiltonian terms and a tunneling term. The Dirac Hamiltonian for a rotated layer is given, and the continuum model for the tunneling term is derived by assuming that the inter-layer tunneling amplitude is a smooth function of separation. A π-band tight-binding calculation yields a continuum limit where tunneling is local. The authors derive a continuum model for the tunneling term and find that the amplitude for an electron to hop between sublattices is given by a specific expression. By comparing with the known electronic structure of an AB-stacked bilayer, they set the tunneling amplitude to a specific value. The spectrum is independent of the separation d for θ ≠ 0. The continuum model captures the local stacking sequence of the misaligned layers, and the periodicity is controlled by reciprocal lattice vectors. The authors solve numerically for the moiré bands using the plane-wave expansion and find that the velocity at the Dirac point oscillates with the twist angle, vanishing at a series of magic angles. These angles give rise to large densities-of-states and large counterflow conductivities. The authors also find that the Dirac-point velocity vanishes at θ ≈ 1.05°, and that the vanishing velocity is accompanied by