A bstract Measurements of CP observables in B 0 → DK ∗0 decays are presented, where D represents a superposition of D 0 and $$ {\overline{D}}^0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mn>0</mml:mn> </mml:msup> </mml:math> states. The D meson is reconstructed in the two-body final states K + π − , π + K − , K + K − and π + π − , and, for the first time, in the fourbody final states K + π − π + π − , π + K − π + π − and π + π − π + π − . The analysis uses a sample of neutral B mesons produced in proton-proton collisions, corresponding to an integrated luminosity of 1.0, 2.0 and 1.8 fb −1 collected with the LHCb detector at centre-of-mass energies of $$ \sqrt{s}=7,8 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>s</mml:mi> </mml:msqrt> <mml:mo>=</mml:mo> <mml:mn>7</mml:mn> <mml:mo>,</mml:mo> <mml:mn>8</mml:mn> </mml:math> and 13 TeV, respectively. First observations of the decays B 0 → D ( π + K − ) K ∗0 and B 0 → D ( π + π − π + π − ) K ∗0 are obtained. The measured observables are interpreted in terms of the CP -violating weak phase γ .