  The final results of the search for the lepton flavour violating decay
$\mu^{+} \rightarrow {\rm e^{+}} \gamma$ based on the full dataset collected by
the MEG experiment at the Paul Scherrer Institut in the period 2009--2013 and
totalling $7.5\times 10^{14}$ stopped muons on target are presented. No
significant excess of events is observed in the dataset with respect to the
expected background and a new upper limit on the branching ratio of this decay
of $BR( \mu^{+} \rightarrow {\rm e^{+}} \gamma ) < 4.2 \times 10^{-13}$ (90\%\
confidence level) is established, which represents the most stringent limit on
the existence of this decay to date.
