The streamline patterns of some simple two-dimensional Stokes flows are studied and the results used both to understand and to predict the streamlines of flows in more complicated geometries, in particular the streamlines of flows that contain eddies or regions of closed streamlines. Initially, streamline patterns are studied locally, either around some special point, such as a stagnation point or a point where a streamline meets a wall, or in a special region, such as a corner. The use of these local analyses is illustrated by finding the streamlines for shear flow around a rotating cylinder; the illustration also shows how fluid in Stokes flow can be turned back on itself (‘blocked’). The local analyses of flow in a corner are used to understand the eddy patterns that have been discovered in a variety of flows. The eddies occur in corner-like regions and in these regions the flow can be regarded as the superposition of two components. One component, the eddy flow, is the result of flow outside the corner stirring the fluid in the corner, while the other is the direct result of local conditions in the corner. The competition between these two components determines whether eddies actually appear in a given flow. Finally, the approach developed here is applied to a new flow situation, namely a shear flow which is bounded by a moving wall and which contains a stationary cylinder touching the wall. The streamlines deduced for different ratios of the shear strength to the wall velocity show both new eddy patterns and unexpected regions of blocked flow.