In this paper, we investigate the Möbius geometry of curves in ${\mathbb{R}}^4$. First, using moving frame methods we construct a complete system of Möbius invariants for regular curves in ${\mathbb{R}}^4$ by the isometric invariants. Second, we completely classify the Möbius homogeneous curves in ${\mathbb{R}}^4$ up to a Möbius transformation of ${\mathbb{R}}^4$.

Let $f:M^n\to {\mathbb{R}}_1^{n+1}$ be an $n$-dimensional umbilic-free spacelike hypersurface in the $(n+1)$-dimensional Lorentzian space ${\mathbb{R}}_1^{n+1}$ with an induced metric $I$. Let $II$ be the second fundamental form and $H$ the mean curvature of $f$. One can define the conformal metric $g=\frac{n}{n-1}(\Vert II{\Vert }^2-n{H}^2)I$ on $f(M^n)$, which is invariant under the conformal transformation group of ${\mathbb{R}}_1^{n+1}$. If the Ricci curvature of $g$ is constant, then the spacelike hypersurface $f$ is called a conformal Einstein hypersurface. In this paper, we completely classify the $n$-dimensional spacelike conformal Einstein hypersurfaces up to a conformal transformation of ${\mathbb{R}}_1^{n+1}$.