  The $N=Z=36$ nucleus $^{72}$Kr has been studied by inelastic scattering at
intermediate energies. Two targets, $^{9}$Be and $^{197}$Au, were used to
extract the nuclear deformation length, $\delta_\text{N}$, and the reduced $E2$
transition probability, $B(E2)$. The previously unknown non-yrast $2^+$ and
$4^+$ states as well as a new candidate for the octupole $3^-$ state have been
observed in the scattering on the Be target and placed in the level scheme
based on $\gamma-\gamma$ coincidences. The second $2^+$ state was also observed
in the scattering on the Au target and the $B(E2;\;2^+_2 \rightarrow 0^+_1)$
value could be determined for the first time. Analyzing the results in terms of
a two-band mixing model shows clear evidence for a oblate-prolate shape
coexistence and can be explained by a shape change from an oblate ground state
to prolate deformed yrast band from the first $2^+$ state. This interpretation
is corroborated by beyond mean field calculations using the Gogny D1S
interaction.
