The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the 3D Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrodinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton (shape invariant) solution which generalizes the text-book one, that is valid in the 1D limit (1D GPE). We investigate stability and collective oscillations of this Bose-condensed bright soliton. By comparing the results of the 3D GPE with those of the 1D GPE, we find a good agreement only for weakly-interacting bright solitons. In particular, contrary to the 1D case, the 3D bright soliton exists only below a critical attractive interaction which depends on the extent of confinement. Analyzing the macroscopic quantum tunneling of the bright soliton on a Gaussian barrier we find that its interference in the tunneling region is strongly suppressed with respect to non-solitonic case, moreover the reflected and transmitted matter waves are not solitonic. Finally, we show that the collapse of the soliton is induced by the scattering on the barrier or by the collision with another matter wave when the density reaches a critical value, for which we derive an accurate analytical formula.