Summary: In this paper, we would like to make a report on Morse inequalities and embeddings for CR manifolds with transversal CR circle action. The results are contained in [\textit{C.-Y. Hsiao} et al., Math. Z. 284, No. 1--2, 441--468 (2016, Zbl 1352.32013); Asian J. Math. 22, No. 3, 413--450 (2018; Zbl 1398.32002); the first author and \textit{C.-Y. Hsiao}, Ann. Global Anal. Geom. 52, No. 3, 313--340 (2017; Zbl 1378.32024); the first author et al., Int. J. Math. 29, No. 9, Article ID 1850061, 35 p. (2018; Zbl 1400.32022); \textit{C.-Y. Hsiao, X. Li} and \textit{G. Marinescu}, ``Equivariant Kodaira embedding of CR manifolds with circle action'', Mich. Math. J. (to appear), \url{arXiv:1603.08872}]. Furthermore, in the last section we will give an explicit example for \(S^1\)-equivariant Szegő kernels and their expansion on the sphere \(S^3\) in \(\mathbb{C}^2\) with respect to a family of transversal CR \(S^1\)-actions.