Charged lepton flavour violating processes are highly suppressed in the standard model, but they are predicted to be enhanced in several new physics extensions including supersymmetry and models with leptoquarks or compositeness. Data collected with the BaBar detector at the SLAC PEP-II e$^+$e$^-$ asymmetric collider at a centre-of-mass energy of 10.36~GeV were used to search for electron-muon flavor violation in $\Upsilon (3S)\rightarrow e^{\pm}\mu^{\pm}$ decays. The search was conducted using a data sample in which 118 million $\Upsilon (3S)$ mesons were produced, corresponding to an integrated luminosity of 27~$fb^{-1}$. There is no evidence of a signal in the $\Upsilon (3S)$ data and we report our results as upper limits on $\mathcal{B}(\Upsilon(3S)\rightarrow e^{\pm}\mu^{\mp})<3.6\times10^{-7}\mathrm{at~ 90\%~ CL}$. / Graduate