The purpose of this paper is to survey fundamental results related to the following problems: 1. Find a necessary and sufficient condition for an isotropic immersion of \(M\) to be totally umbilic in an arbitrary Riemannian manifold \(\widetilde M\). 2. Give a geometric meaning of a constant isotropic immersion of \(M\) into an arbitrary Riemannian manifold \(\widetilde M\). 3. Classify isotropic immersions of \(M^n\) with low codimension \(p\) into a real space form \({\widetilde M}^{n+p}(c;\mathbb R)\). 4. Give a sufficient condition that an isotropic immersion of a rank one symmetric space \(M\) into a real space form \({\widetilde M}(c;\mathbb R)\) has parallel second fundamental form. The author gives an answer to Problem 1 in terms of shape operators of \(M\) in the ambient space \(\widetilde M\) and an answer to Problem 2 by studying circles on the submanifold \(M\). He solves Problem 3 in case of \(p\leq \frac{n+2}{2}\) and Problem 4 by using inequalities related to mean curvatures of these submanifolds. The paper contains some open problems on isotropic immersions.