The notions of metric and affine connection play a fundamental role in classical differential geometry. The first one allows us to measure distances, whereas the second allows us to define the concept of parallelism (more precisely, parallel transport of vectors).

Riemannian geometry is based on the assumption that the way we measure distances determines the way we define the concept of parallelism. Namely, the connection coefficients are assumed to be expressed via the components of the metric tensor in accordance with a certain explicit formula. Such a connection is called the Levi-Civita connection, and such connection coefficients are called Christoffel symbols. The Levi-Civita connection can be invariantly characterised as the (unique) metric compatible connection with zero torsion.

Non-Riemannian geometry is based on the assumption that the way we measure distances is unrelated to the way we define the concept of parallelism. Namely, there is no a priori relationship between the connection and the metric. In particular, the connection may not be metric compatible and (or) may have nonzero torsion.

We suggest a field theory [1] based on non-Riemannian geometry. We choose an arbitrary Lorentz-invariant quadratic action and vary it independently with respect to the metric and the connection. We show that the solutions of the resulting field equations are perfectly sensible. Einstein spaces turn out to be solutions, so the theory reproduces the main effects of General Relativity. Another solution is an explicitly constructed wave of torsion in Minkowski space; this solution may be interpreted as a model for an elementary particle.

[1] D.Vassiliev, Pseudoinstantons in Metric-Affine Field Theory, "General Relativity and Gravitation" 2002 vol. 34 p. 1239-1265.