  The upcoming Mu3e experiment aims to search for the lepton flavour violating
decay $\mu^+ \to e^+ e^- e^+$ with an unprecedented final sensitivity of one
signal decay in $10^{16}$ observed muon decays by making use of an innovative
experimental design based on novel ultra-thin silicon pixel sensors. In a first
phase, the experiment is operated at an existing muon beam line with rates of
up to $10^8$ muons per second. Detailed simulation studies confirm the
feasibility of background-free operation and project single event sensitivities
in the order of $10^{-15}$ for signal decays modelled in an effective field
theory approach. The precise tracking of the decay electrons and large
geometric and momentum acceptance of Mu3e enable searches for physics beyond
the Standard Model in further signatures. Examples of which are searches for
lepton flavour violating two-body decays of the muon into an electron and an
undetected boson as well as for electron-positron resonances in $\mu^+ \to e^+
e^- e^+ \bar{\nu}_\mu \nu_e$ which could result for instance from a dark photon
decay. The Mu3e experiment is expected to be competitive in all of these
channels already in phase I.
