The Mu3e experiment will search for the charged lepton flavor violating decay $\mu^+\rightarrow e^+e^-e^+$ with a targeted branching ratio sensitivity of $10^{-16}$. The sensitivity goal drives the experimental design: To distinguish the neutrinoless signal decay from background processes, excellent momentum, vertex and time resolutions of the detector system are required. An ultra-thin silicon pixel tracking detector will be constructed, complemented by a scintillating fiber and a scintillating tile detector which add precise time information to the tracks. To conduct the experiment within a reasonable time, the detector will have to cope with electrons and positrons originating from muon decays at rates up to $10^9$ per second. The trigger-less readout system collects the detector data, reconstructs tracks and selects events online for offline analysis. This paper describes the current status of design and construction of the Mu3e experiment and emphasizes the progress made in the development of the readout system.