The basics of submanifolds in complex space forms and Sasakian space forms are recalled, and then Chen-type inequalities for different submanifolds in complex and Sasakian space forms are presented. The most important Chen inequalities in real space forms are stated. We give a general construction method for purely real submanifolds and present geometric inequalities for purely submanifolds in complex space forms. We obtain an improved Chen-Ricci inequality for Kaehlerian slant submanifolds in complex space forms. Works on DDVV conjecture are also presented. Next, results on submanifolds in Sasakian manifolds are presented. We prove Chen’s first inequality for contact slant submanifolds in Sasakian space forms. We define Chen-type Sasakian invariants, obtain sharp inequalities for these invariants, and derive characterizations of the equality case in terms of the shape operator. We generalize a result of Chen and obtain a Chen-Ricci inequality for purely real submanifolds with T parallel with respect to the Levi-Civita connection. Another subsection presents certain results for submanifolds in space forms with semi-symmetric metric (respectively, nonmetric) connections. We study statistical submanifolds and their behavior in statistical manifolds of constant curvature. Next, we present results on warped product submanifolds in complex space forms, generalized complex space forms, and quaternion space forms. After that, a new characterization of Einstein spaces by using their curvatures symmetries is given. This chapter represents a collection of results from the author’s papers on this topic; the proofs are given in detail, so the reader can follow the techniques.