The dynamics and the three-dimensional structure of vortices in a linearly stratified, non-rotating fluid are investigated by means of laboratory experiments, an analytical model and through numerical simulations. The laboratory experiments show that such vortices have a thin pancake-like appearance. Due to vertical diffusion of momentum the strength of these vortices decreases rapidly and their thickness increases in time. Also it is found that inside a vortex the linear ambient density profile becomes perturbed, resulting in a local steepening of the density gradient. Based on the assumption of a quasi-two-dimensional axisymmetric flow (i.e. with zero vertical velocity) a model is derived from the Boussinesq equations that illustrates that the velocity field of the vortex decays due to diffusion and that the vortex is in so-called cyclostrophic balance. This means that the centrifugal force inside the vortex is balanced by a pressure gradient force that is provided by a perturbation of the density profile in a way that is observed in the experiments. Numerical simulations are performed, using a finite difference method in a cylindrical coordinate system. As an initial condition the three-dimensional vorticity and density structure of the vortex, found with the diffusion model, are used. The influence of the Froude number, Schmidt number and Reynolds number, as well as the initial thickness of the vortex, on the evolution of the flow are investigated. For a specific combination of flow parameters it is found that during the decay of the vortex the relaxation of the isopycnals back to their undisturbed positions can result in a stretching of the vortex. Potential energy of the perturbed isopycnals is then converted into kinetic energy of the vortex. However, when the stratification is strong enough (i.e. for small Froude numbers), the evolution of the vortex can be described almost perfectly by the diffusion model alone.