This book is the first volume in a two-volume set, which will provide the complete proof of classification of two important classes of geometries, closely related to each other: Petersen and tilde geometries. There is an infinite family of tilde geometries associated with nonsplit extensions of symplectic groups over a field of two elements. Besides that there are twelve exceptional Petersen and tilde geometries. These exceptional geometries are related to sporadic simple groups, including the famous Monster group and this volume gives a construction for each of the Petersen and tilde geometries that provides an independent existence proof for the corresponding automorphism group. Important applications of Petersen and tilde geometries are considered, including the so-called Y-presentations for the Monster and related groups, and a complete identification of Y-groups is given. This is an essential purchase for researchers in finite group theory, finite geometries and algebraic combinatorics.