The development and structure of flows in eccentric annular channels and their dependence on inlet conditions, inner-to-outer diameter ratio $d/D$, eccentricity $e=2{\rm\Delta}y/(D-d)$, where ${\rm\Delta}y$ is the distance between the axes of the inner and outer cylinders, and Reynolds number $\mathit{Re}$, based on the hydraulic diameter and the bulk velocity, were studied experimentally, with focus on the phenomena of gap instability and the resulting vortex street. Experimental conditions covered a Reynolds number range between 0 and 19 000, an eccentricity range from 0 to 0.9 and inner-to-outer diameter ratios equal to 0.25, 0.50 and 0.75. Much of the discussion is based on measurements in the middle of the narrow annular gap, where the phenomena of interest could be observed most vividly. In the range $\mathit{Re}<10\,000$, the Strouhal number, the normalized mid-gap axial flow velocity and the normalized axial and cross-flow fluctuations at mid-gap were found to increase with increasing $\mathit{Re}$ and to depend strongly on inlet conditions. At higher Reynolds numbers, however, these parameters reached asymptotic values that were less sensitive to inlet conditions. We constructed a map for the various stages of periodic motions versus eccentricity and Reynolds number and found that for $e<0.5$ or $\mathit{Re}<1100$ the flow was unconditionally stable, as far as gap instability is concerned. For $e\leqslant 0.5$, transition to turbulence occurred at $\mathit{Re}\approx 6000$, whereas, for $0.6\leqslant e\leqslant 0.9$, the critical Reynolds number for the formation of periodic motions was found to increase with eccentricity from 1100 for $e=0.6$ to 3800 for $e=0.9$.