The aim of this book is to define and study the $({\cal{T}}_{\!a}, {\cal{T}}_{\!b})$-pseudosymmetric, $({\cal{T}}_{\!a}, {\cal{T}}_{\!b},S^{\ell})$-pseudosymmetric, $({\cal{T}}_{\!a},S_{{\cal{T}}_{b}})$-pseudosymmetric and $({\cal{T}}_{\!a},S_{{\cal{T}}_{\!b}},S^{\ell})$-pseudosymmetric semi-Riemannian manifolds. Then study these type of pseudosymmetry for $(N(K),\xi)$-semi-Riemannian manifolds.