The authors first explain a general method for investigating the natural differential operators, which is based on the use of an auxiliary linear connection. Using this method, they first rededuce a classical result of R. S. Palais [Trans. Am. Math. Soc. 92, 125-141 (1959; Zbl 0092.308)] that every natural linear operator from \(\Lambda^ pT^*M\) into \(\Lambda^{p+1}T^*M\) is a constant multiple of the exterior differential. Then the authors prove that every finite order natural bilinear operator from \(TM\times_ MTM\) into TM is a constant multiple of the Lie bracket. The last result is that the unique first order natural operator from the bundle of all indefinite Riemannian metrics into the bundle of all linear connections is the Levi-Civita connection.