In this paper, we study Möbius characterizations of submanifolds without umbilical points in a unit sphere $S^{n+p}(1)$ . First of all, we proved that, for an $n$ -dimensional $(n\geq 2)$ submanifold $\mathbf x:M\mapsto S^{n+p}(1)$ without umbilical points and with vanishing Möbius form $\Phi$ , if
$(n-2)||\tilde{\mathbf{A}}|| \leq\sqrt{\frac{n-1}n} \left\{ nR-\frac{1}{n}[(n-1)\left( 2-\frac{1}{p} \right)-1] \right\}$ is satisfied, then, $\mathbf x$ is Möbius equivalent to an open part of either the Riemannian product $S^{n-1}(r)\times S^{1}\left(\sqrt{1-r^2}\right)$ in $S^{n+1}(1)$ , or the image of the conformal diffeomorphism $\sigma$ of the standard cylinder $S^{n-1}(1)\times \mathbf{R}$ in $\mathbf{R}^{n+1}$ , or the image of the conformal diffeomorphism $\tau$ of the Riemannian product $S^{n-1}(r)\times \mathbf{H}^{1}\left(\sqrt{1+r^2}\right)$ in $\mathbf{H}^{n+1}$ , or $\mathbf x$ is locally Möbius equivalent to the Veronese surface in $S^4(1)$ . When $p=1$ , our pinching condition is the same as in Main Theorem of Hu and Li [6], in which they assumed that $M$ is compact and the Möbius scalar curvature $n(n-1)R$ is constant. Secondly, we consider the Möbius sectional curvature of the immersion $\mathbf x$ . We obtained that, for an $n$ -dimensional compact submanifold $\mathbf x:M\mapsto S^{n+p}(1)$ without umbilical points and with vanishing form $\Phi$ , if the Möbius scalar curvature $n(n-1)R$ of the immersion $\mathbf x$ is constant and the Möbius sectional curvature $K$ of the immersion $\mathbf x$ satisfies $K\geq 0$ when $p=1$ and $K>0$ when $p>1$ . Then, $\mathbf x$ is Möbius equivalent to either the Riemannian product $S^k(r)\times S^{n-k}\left(\sqrt{1-r^2}\right)$ , for $k=1, 2, \cdots, n-1$ , in $S^{n+1}(1)$ ; or $\mathbf x$ is Möbius equivalent to a compact minimal submanifold with constant scalar curvature in $S^{n+p}(1)$ .