  We investigate the complex relationship between the growth conditions and the
structural and transport properties of $\mathrm{Ti_xO_y}$ thin films grown by
molecular beam epitaxy. Transport properties ranging from metallicity to
superconductivity and insulating states are stabilized by effectively tuning
the O/Ti ratio via the Ti flux rate and the O partial pressure, $P_{Ox}$, for
films grown on (0001)-$\mathrm{Al_2 O_3}$ substrates at 850$^{\circ}$ C. A
cubic $c-\mathrm{TiO_{1\pm\delta}}$ buffer layer is formed for low O/Ti ratios
while a corundum cr-$\mathrm{Ti_2 O_3}$ layer is formed under higher oxidizing
conditions. Metallicity is observed for c-$\mathrm{TiO_{1-\delta}}$ buffer
layers. The superconducting $\mathrm{\gamma-Ti_3 O_5}$ Magn\'eli phase is found
to nucleate on a c-$\mathrm{TiO_{1-\delta}}$ buffer for intermediate $P_{Ox}$
conditions and an insulator-superconducting transition is observed at 4.5 K
(T$_C^{onset}=6 K$) for 85 nm thick films. Strain relaxation of the
$\mathrm{\gamma-Ti_3 O_5}$ occurs with increasing film thickness and correlates
with a thickness-dependent increase in T$_C$ observed for $\mathrm{Ti_xO_y}$
thin films.
