The aim of the paper is to construct a calculus of pseudodifferential operators (PDOs) on a smooth manifold $M$ without using local coordinate systems. Instead we deal with linear connections $\Gamma$ of $M$.

The fact that a linear connection $\Gamma$ is a global object enables one to associate with a PDO its full symbol, which is a function on the cotangent bundle $T^*M$ (depending on the choice of $\Gamma\,$). This idea was put forward by H. Widom. He has suggested a method of defining full symbols of PDOs on a manifold with a linear connection and constructed a version of symbolic calculus. However, H. Widom defined the classes of PDOs in local coordinates, using standard local phase functions.

On the contrary, in the paper the PDOs are defined in a coordinate-free way, with use of invariant oscillatory integrals over $T^*M\,$. The invariant approach allows us to define $\tau$-symbols of pseudodifferential operators acting on a manifold; in particular, we introduce the Weyl symbols. In fact, it is a kind of quantization on a manifold provided with a linear connection.

The results are applied to study of semi-elliptic differential operators and functions of the Laplacian on a closed Riemannian manifold.

1991 Mathematical Subject Classification: 35S05, 58G30.