In this chapter, linear dilations in abstract (Banach and Hilbert) spaces are introduced as a group of linear bounded operatorsGroup of linear bounded operatorsLinear bounded operatorOperator satisfying a limit propertyLimit property. Special attention is paid to strongly continuous linear dilations in Lebesgue and Sobolev spacesSobolev space. Geometric objects induced by dilations are considered. The canonical homogeneous normCanonical homogeneous norm and the canonical homogeneous inner productInner product are defined for generalized homogeneous Banach and Hilbert spacesHilbert space, respectively.