The study of quasi-two-dimensional turbulent jets is relevant to chemical reactors, the coking process in oil refinement, as well as rivers flowing into lakes or oceans. In the event of a spillage of pollutants into a river, it is critical to understand how these agents disperse with the flow in order to assess damage to the environment. For such flows, characteristic streamwise and cross-stream dimensions can be much larger than the fluid-layer thickness, and so the flow develops in a confined environment. When the distance away from the discharge location is larger than ten times the fluid-layer thickness, the flow is referred to as a quasi-two-dimensional jet. From experimental observations using dyed jets and particle image velocimetry, we find that the structure of a quasi-two-dimensional jet consists of a high-speed meandering core with large counter-rotating eddies developing on alternate sides of the core. The core and eddy structure is self-similar with distance from the discharge location. The Gaussianity of the cross-stream distribution of the time-averaged velocity is due, in part, to the sinuous instability of the core. To understand the transport and dispersion properties of quasi-two-dimensional jets we use a time-dependent advection--diffusion equation, with a mixing length hypothesis accounting for the turbulent eddy diffusivity. The model is supported by experimental releases of dye in jets or numerical releases of virtual passive tracers in experimentally-measured jet velocity fields. We consider the statistical properties of this flow by releasing and then tracking large clusters of virtual particles in the jet velocity field. The probability distributions of two-point properties (such as the distance between two particles) reveal large streamwise dispersion. Owing to this streamwise dispersive effect, a significant amount of tracers can be transported faster than the speed predicted by a simple advection model. Using potential theory, we determine the flow induced by a quasi-two-dimensional jet confined in a rectangular domain. The streamlines of the induced flow predicted by the theory agree with experimental measurements away from the jet boundary. Finally, we investigate the case of a quasi-two-dimensional particle-laden jet. Depending on the bulk concentration of dense particles, we identify different flow regimes. At low concentrations, the jet features the same core and eddy structure observed without the particles, and thus quasi-two-dimensional jet theory can apply to some extent. At larger concentrations, we observe an oscillating instability of the particle-laden jet.