  The recent LHCb measurement of $R_{K^*}$ in two $q^2$ bins, when combined
with the earlier measurement of $R_K$, strongly suggests lepton flavour
non-universal new physics in semi-leptonic $B$ meson decays. Motivated by these
intriguing hints of new physics, several authors have considered vector, axial
vector, scalar and pseudo scalar operators as possible explanations of these
measurements. However, tensor operators have widely been neglected in this
context. In this paper, we consider the effect of tensor operators in $R_K$ and
$R_{K^*}$. We find that, unlike other local operators, tensor operators can
comfortably produce both of $R_{K^*} ^{\rm low}$ and $R_{K^*} ^{\rm central}$
close to their experimental central values. However, a simultaneous explanation
of $R_K$ is not possible with only Tensor operators, and other vector or axial
vector operators are needed. In fact, we find that combination of vector and
tensor operators can provide simultaneous explanations of all the anomalies
comfortably at the $1 \sigma$ level, a scenario which is hard to achieve with
only vector or axial vector operators. We also comment on the compatibility of
the various new physics solutions with the measurements of the inclusive decay
$B_d \to X_s \ell^+ \ell^-$.
