This survey consists of two major parts. In Part I, after background material, we classify all nontrivial analytic CR manifolds M of dimensions 3, 4, 5 at a generic point into six pairwise nonequivalent classes, which we designate by \(\textsf { I}\) , \(\textsf{ II}\) , \(\textsf{ III}_\textsf {1}\) , \(\textsf{ III}_\textsf {2}\) , \(\textsf{ IV}_\textsf {1}\) , \(\textsf {IV}_\textsf {2}\) . This classification is based on the Lie-bracket structure of the CR-invariant bundles \(T^{1,0} M\) and \(T^{0,1}M\) . We also present in Part I adapted partial normal forms for the defining equations of the concerned CR manifolds. In Part II, we review existing results on the biholomorphic equivalence problem of the introduced classes along with relevant topics on their complete normal forms, Cartan geometries and CR symmetry groups.
