A theoretical study of persistent currents in liquid
He4 is given, in which a system of
N interacting bosons confined to a cylindrical container is chosen as a model. It is argued that these currents are metastable and a general model-independent stability criterion valid for all temperatures is derived. In this formulation it is essential that there is a single-particle state ${\ensuremath{\phi}}_{c}$ with nonzero angular momentum
lc which is occupied by a macroscopic number [
O(N)] of particles. The stability criterion is applied to the study of two soluble models, one, an independent-particle model studied for all temperatures, the other a quasiparticle model studied for
T=0\ifmmode^\circ\else\textdegree\fi{}K. For both models persistent currents are possible at
T=0 provided that the interactions between particles are sufficiently strong and
lc does not exceed a characteristic critical value
lM. The quantity
lM is calculated for different choices of single-particle functions, and in particular for the solutions of a Hartree equation, which is relevant to a description of line vortices with quantized circulation. The density ${\ensuremath{\rho}}_{c}(T)$ of particles macroscopically occupying ${\ensuremath{\phi}}_{c}$ at a temperature
T, as well as the critical temperature
T∗ above which no persistent currents are possible, are calculated for the independent-particle model.