In this paper, we present local stereological estimators of Minkowski tensors defined on convex bodies in $\mathbb{R}^d$. Special cases cover a number of well-known local stereological estimators of volume and surface area in $\mathbb{R}^3$, but the general set-up also provides new local stereological estimators of various types of centres of gravity and tensors of rank two. Rank two tensors can be represented as ellipsoids and contain information about shape and orientation. The performance of some of the estimators of centres of gravity and volume tensors of rank two is investigated by simulation. Keywords: Ellipsoidal approximation, Local stereology, Minkowski tensors, Particle shape, Particle orientation, Rotational integral geometry