A theory of quantized vortex excitations in boson systems is presented. For weakly repelling bosons, the vortex state is characterized by a finite fraction of particles in a single-particle state of integer angular momentum. The radial dependence of the highly occupied state is determined by a self-consistent field equation. The radial function and particle density are nearly constant except within a core, where they drop to zero. The core size is related to the de Broglie wavelength associated with the mean interaction energy per particle. The expectation value of the velocity has a classical vortex radial dependence. In the Hartree approximation, vorticity is zero everywhere except on the vortex line. When zero-point oscillations of the phonon field are included, the vorticity is spread over the core. These results confirm the intuitive arguments of Onsager and Feynman. Phonons moving perpendicular to the vortex line are coherent excitations with equal and opposite angular momentum relative to the moving particle substratum. The vortex motion resolves the degeneracy of Bogoljubov phonons with respect to the azimuthal quantum number. The idea that liquid helium allows macroscopic vortex motions, similar to ordinary liquids, has been crucial in interpreting recent experiments. Experiments by VINEN provide evidence for free vortex lines with quantized circulation. The presence of vortex motions superimposed on a phonon field suggests that quantized vortex lines explain the low critical velocity. The quantum mechanical description of a vortex has puzzling aspects, with arguments for and against the existence of vorticity. The wave function suggests a velocity potential, implying no vorticity, but this has not been conclusively shown. Onsager and Feynman argue that vorticity may exist in concentrated regions. Their reasoning indicates that quantized vortex lines may exist, supporting the explanation of superfluid phenomena.A theory of quantized vortex excitations in boson systems is presented. For weakly repelling bosons, the vortex state is characterized by a finite fraction of particles in a single-particle state of integer angular momentum. The radial dependence of the highly occupied state is determined by a self-consistent field equation. The radial function and particle density are nearly constant except within a core, where they drop to zero. The core size is related to the de Broglie wavelength associated with the mean interaction energy per particle. The expectation value of the velocity has a classical vortex radial dependence. In the Hartree approximation, vorticity is zero everywhere except on the vortex line. When zero-point oscillations of the phonon field are included, the vorticity is spread over the core. These results confirm the intuitive arguments of Onsager and Feynman. Phonons moving perpendicular to the vortex line are coherent excitations with equal and opposite angular momentum relative to the moving particle substratum. The vortex motion resolves the degeneracy of Bogoljubov phonons with respect to the azimuthal quantum number. The idea that liquid helium allows macroscopic vortex motions, similar to ordinary liquids, has been crucial in interpreting recent experiments. Experiments by VINEN provide evidence for free vortex lines with quantized circulation. The presence of vortex motions superimposed on a phonon field suggests that quantized vortex lines explain the low critical velocity. The quantum mechanical description of a vortex has puzzling aspects, with arguments for and against the existence of vorticity. The wave function suggests a velocity potential, implying no vorticity, but this has not been conclusively shown. Onsager and Feynman argue that vorticity may exist in concentrated regions. Their reasoning indicates that quantized vortex lines may exist, supporting the explanation of superfluid phenomena.