The interaction of a cylindrical converging shock wave with an initially perturbed gaseous interface is studied experimentally. The cylindrical converging shock is generated in an ordinary shock tube but with a specially designed test section, in which the incident planar shock wave is directly converted into a cylindrical one. Two kinds of typical initial interfaces involving gas bubble and gas cylinder are employed. A high-speed video camera combined with schlieren or planar Mie scattering photography is utilized to capture the evolution process of flow structures. The distribution of baroclinic vorticity on the interface induced by the cylindrical shock and the reflected shock from the center of convergence results in distinct phenomena. In the gas bubble case, the shock focusing and the jet formation are observed and the turbulent mixing of two fluids is promoted because of the gradually changed shock strength and complex shock structures in the converging part. In the gas cylinder case, a counter-rotating vortex pair is formed after the impact of the converging shock and its rotating direction may be changed when interacting with the reflected shock for a relatively long reflection distance. The variations of the interface displacements and structural dimensions with time are further measured. It is found that these quantities are different from those in the planar counterpart because of the shock curvature, the Mach number effect and the complex shock reflection within the converging shock tube test section. Therefore, the experiments reported here exhibit the great potential of this experimental method in study of the Richtmyer-Meshkov instability induced by converging shock waves.