  Laguerre geometry of surfaces in $\R^3$ is given in the book of Blaschke [1],
and have been studied by E.Musso and L.Nicolodi [5], [6], [7], B. Palmer [8]
and other authors. In this paper we study Laguerre differential geometry of
hypersurfaces in $\R^n$. For any umbilical free hypersurface $x: M\to\R^n$ with
non-zero principal curvatures we define a Laguerre invariant metric $g$ on $M$
and a Laguerre invariant self-adjoint operator ${\mathbb S}: TM\to TM$, and
show that $\{g,{\mathbb S}\}$ is a complete Laguerre invariant system for
hypersurfaces in $\R^n$ with $n\ge 4$. We calculate the Euler-Lagrange equation
for the Laguerre volume functional of Laguerre metric by using Laguerre
invariants. Using the Euclidean space $\R^n$, the Lorentzian space $\R^n_1$ and
the degenerate space $\R^n_0$ we define three Laguerre space forms $U\R^n$,
$U\R^n_1$ and $U\R^n_0$ and define the Laguerre embedding $ U\R^n_1\to U\R^n$
and $U\R^n_0\to U\R^n$, analogue to the Moebius geometry where we have Moebius
space forms $S^n$, $\H^n$ and $\R^n$ (spaces of constant curvature) and
conformal embedding $\H^n\to S^n$ and $\R^n\to S^n$ (cf. [4], [10]). Using
these Laguerre embedding we can unify the Laguerre geometry of hypersurfaces in
$\R^n$, $\R^n_1$ and $\R^n_0$. As an example we show that minimal surfaces in
$\R^3_1$ or $\R_0^3$ are Laguerre minimal in $\R^3$.
