The spin-up from rest of (i) a homogeneous and (ii) a linearly stratified fluid in a rectangular container has been examined in the laboratory. In the spin-up process leading to the ultimate state of rigid-body rotation, three main stages can be discerned, these being (1) the starting flow, characterized by zero absolute vorticity, (2) flow separation due to cyclonic vorticity generation at the lateral tank walls, and (3) a subsequent organization of the flow into a regular array of alternately cyclonic and anticyclonic cells. During the final stage the flow in these cells gradually decays due to the spin-down/spin-up mechanism provided by the Ekman boundary layer present at the bottom of each cell. Experiments have been performed with free-surface and rigid-lid upper boundary conditions, and the organization of the flow in these cases was observed to be essentially different. In particular, it was noted that the central cell in the free-surface case is always cyclonic. A model for this behavior is advanced, in terms of the tendency of cyclonic vortices to move toward the rotation axis in the free-surface configuration.