We will discuss some natural linear differential operators for different geometric structures. For a Riemannian manifold of dimension n, an interesting family consist of operators of form $S^*S$, where $S^*$ is the operator formally adjoint to $S$ and where $S$ is the the gradient in the sense of Stein and Weiss, i.e., $S$ is an $O(n)$-irreducible summand of the covariant derivative. We will discuss the ellipticity and the boundary properties such operators. In particular, we will discuss natural boundary conditions for the elliptic operators and the ellipticity of these conditions at the boundary. One of the consequences of such the ellipticity for a given boundary condition is the existence of a basis for $L^2$ composed of smooth sections that are the eigenvectors of the operator and satisfy the boundary condition.

We will also discuss the Laplace type operators of form $\mathop{\rm div} \mathop{\rm grad}$ acting in tensor bundles on a Riemannian or symplectic manifold. Here the operator grad is a natural generalization of the classic gradient operator acting on vector fields. The negative divergence $-\mathop{\rm div}$ is the operator formally adjoint to $\mathop{\rm grad}$. The second order operator $-\mathop{\rm div} \mathop{\rm grad}$ relates to the Lichnerowicz Laplacian which acts on tensors (forms) of any symmetry. The relation involves the curvature. We will also mention the problem of restriction of differential operators (so the Stein-Weiss gradients in particular) to submanifolds or to the leaves of a foliation.