The paper presents a quantitative continuum theory of "flocking," the collective coherent motion of large numbers of self-propelled organisms. The model predicts the existence of an "ordered phase" where all members of a flock move together with a mean velocity, demonstrating spontaneous symmetry breaking. The authors analyze the model to make detailed predictions for the long-distance, long-time behavior of the "broken symmetry state." They identify "Goldstone modes" associated with the spontaneous rotational symmetry breaking, which mix with modes related to conservation of bird number to produce propagating sound modes. These sound modes lead to significant fluctuations in flock density, larger than those in equilibrium systems. The model is similar to the Navier-Stokes equations for compressible fluids and a relaxational time-dependent Ginsburg-Landau theory for isotropic ferromagnets. In spatial dimensions \(d > 4\), the long-distance behavior is described by a linearized theory, while in \(d < 4\), non-linear fluctuation effects alter the behavior. In two dimensions, the flock exhibits a true, long-range ordered, spontaneously broken symmetry state, despite the Mermin-Wagner theorem. The paper also discusses an anisotropic model where birds prefer to move in certain directions and provides exact scaling exponents for all spatial dimensions, including \(d = 3\).The paper presents a quantitative continuum theory of "flocking," the collective coherent motion of large numbers of self-propelled organisms. The model predicts the existence of an "ordered phase" where all members of a flock move together with a mean velocity, demonstrating spontaneous symmetry breaking. The authors analyze the model to make detailed predictions for the long-distance, long-time behavior of the "broken symmetry state." They identify "Goldstone modes" associated with the spontaneous rotational symmetry breaking, which mix with modes related to conservation of bird number to produce propagating sound modes. These sound modes lead to significant fluctuations in flock density, larger than those in equilibrium systems. The model is similar to the Navier-Stokes equations for compressible fluids and a relaxational time-dependent Ginsburg-Landau theory for isotropic ferromagnets. In spatial dimensions \(d > 4\), the long-distance behavior is described by a linearized theory, while in \(d < 4\), non-linear fluctuation effects alter the behavior. In two dimensions, the flock exhibits a true, long-range ordered, spontaneously broken symmetry state, despite the Mermin-Wagner theorem. The paper also discusses an anisotropic model where birds prefer to move in certain directions and provides exact scaling exponents for all spatial dimensions, including \(d = 3\).