We introduce a Wavelength Convertible Flexible Optical WDM (WC-FWDM) network architecture that offers wavelength conversion capability in addition to dynamic allocation of resources as in FWDM networks. We introduce the on-line routing, wavelength assignment and spectrum allocation problem in WC-FWDM networks for the first time, and propose an auxiliary-graph based polynomial-time heuristic-algorithm. Numerical results demonstrate that WC-FWDM networks can significantly reduce the blocking of connections and improve network throughput over FWDM networks with no wavelength conversion capability and fixed grid networks. Furthermore, sparse wavelength conversion can gain the same benefits as full wavelength conversion through wavelength converter sharing.