  Classical (Euclidean) Laguerre geometry studies oriented hyperplanes,
oriented hyperspheres, and their oriented contact in Euclidean space. We
describe how this can be generalized to arbitrary Cayley-Klein spaces, in
particular hyperbolic and elliptic space, and study the corresponding groups of
Laguerre transformations. We give an introduction to Lie geometry and describe
how these Laguerre geometries can be obtained as subgeometries. As an
application of two-dimensional Lie and Laguerre geometry we study the
properties of incircular nets.
