The Mu3e experiment at the Paul-Scherrer-Institut (PSI) searches for the charged lepton flavour violating decay $\mu^+\rightarrow e^+e^-e^+$.
This decay mode is extremely suppressed in the Standard Model, such that any observation would be a clear signature for new physics being at play.
The experiment will be conducted in two phases.
In Phase I, a single event sensitivity of $2 \times 10^{-15}$ is projected to be reached using the Compact Muon Beamline present at PSI.
To reach the ultimate sensitivity of $10^{-16}$ in Phase II, an upgrade of the detector as well as a higher intensity muon beamline will be required.

The detector system has to provide excellent tracking efficiency as well as momentum, vertex and time resolutions to reach the experimental goals.
An unprecedentedly thin silicon pixel tracking detector is being constructed using HV-MAPS, ultra-light services and a gaseous helium cooling system.
It is complemented by timing detectors consisting of scintillating fibres and tiles.
The full detector is placed inside a solenoidal magnetic field of 1 T.

A first run integrating several subdetector prototypes was successfully conducted at PSI in 2021.
A second run using cosmic muons took place in the first half of 2022.
While the design of the final detector components is being completed, the Mu3e experiment is entering the production stage.
Commissioning of the final Phase I detector is planned to start in 2023.
In this publication, the status of the Mu3e experiment is presented.