Exact solutions for finite-amplitude motion of an inviscid, incompressible and stratified fluid corresponding to rectilinear vortices or vortex rings are found. Some of the rectilinear vortex pairs, in circular-cylindrical form, can propagate with a constant horizontal velocity in a surrounding homogeneous fluid, and some of the spherical vortex rings can be placed in a surrounding quiescent homogeneous fluid without translational propagation. Equations governing finite-amplitude two-dimensional motion and axisymmetric motion with swirl in the presence of a magnetic field are derived for a stratified conducting fluid, and the effects of a magnetic field on vortex motion are ascertained.