This paper concerns the stability of the von Kármán swirling flow between coaxial disks. A linear stability analysis shows that for moderate Reynolds numbers ($Re\le50$) and for any rotation ratio $s\in[-1,1[$ there is a radial location $r_{pc}$ from which the self-similar von Kármán solutions become unstable to axisymmetric disturbances. When the disks are moderately counter-rotating ($s\in[-0.56,0[$), two different disturbances (types I and II) appear at the same critical radius. A spatio-temporal analysis shows that, at a very short distance from this critical radius, the first disturbance (type I) becomes absolutely unstable whereas the second (type II) remains convectively unstable. Outside this range of aspect ratios, all the disturbances examined are found to be absolutely unstable. The flow between two coaxial rotating disks enclosed in a stationary sidewall is then numerically investigated. For sufficently large aspect ratios, the cavity flow is found to be globally unstable for axisymmetric disturbances similar to that calculated with the self-similar solutions. The flow in cavities with aspect ratios smaller than $R\,{\approx}\,10.3$ (and $Re\,{\le}\,50$) is not destabilized by these axisymmetric disturbances. An experimental investigation conducted for a cavity with aspect ratio $R\,{=}\,15$ confirms the numerical results. Axisymmetric disturbances similar to those calculated for the same cavity are detected and three-dimensional modes can also be observed near the sidewall.