This lecture notes give a survey of basic facts related to geometry of manifolds endowed with a torsion free connection. We pay the attention especially on geometries which come from the existence on some characteristic torsion free connection closely related to some metric, in general case of an arbitrary signature. So we study in this spirit affine differential geometry, Weyl and Codazzi geometries. One can join naturally these two structures: a torsion free connection and a metric, and the corresponding groups of transformations. We present basic facts related to these groups. To study geometry of manifolds endowed with a torsion free connection, a powerful tool is, of course, the corresponding curvatures. Therefore the curvature is appeared in all five sections of this lecture notes.