Laminar Flow Control is a developing area of research concerned with using various devices and design features to maintain laminar attached flow (typically of air) over the surface of a body. A principal way of achieving this is to place arrays of suction holes at locations on the surface where flow separation or transition towards turbulence is otherwise likely to occur. Suction holes tend to be effective in this role as they remove the slowest moving fluid close to the surface. This leaves a flow that is more stable and more capable of remaining attached under a strong adverse pressure gradient. However, experiments reveal that beneficial effects can only be gained from holes up to a certain size and suction strength. Beyond these limits, the suction begins to destroy the laminar flow entirely. Taking high Reynolds numbers, the author examines the structure of steady laminar flow past a single slot or hole undergoing varying degrees of suction. In all cases, the dimensions of the slot or hole remain fixed, of size either comparable to or smaller than the local boundary-layer thickness. For the case of a two-dimensional suction slot, the fiow response exhibits both beneficial and detrimental behaviour for maintenance of laminar flow depending on the length scale of the slot. In particular, it is shown that shorter slots are able to produce a large-scale region of separated flow which engulfs the downstream edge of the slot and can even persist for a significant distance downstream of the slot. In the case of a three-dimensional suction hole, a complex wake structure is produced at higher suction strengths consisting of one or more pairs of horseshoe-type trailing vortices. A possible mechanism for the generation of these horseshoe vortices is also examined in isolation.