[Author] Oleg Reinov [Title] Some remarks on Approximation Properties with Applications [AMS Subj-class] 46B28 Spaces of operators; tensor products; approximation properties [Abstract] We study some known approximation properties and introduce and investigate several new approximation properties, closely connected with different quasi-normed tensor products. These are the properties like the $AP_s$ or $AP_{(s,w)}$ for $s\in (0,1],$ which give us the possibility to identify the spaces of $s$-nuclear and $(s,w)$-nuclear operators with the corresponding tensor products (e.g., related to Lorentz sequence spaces). Some applications are given (in particular, we present not difficult proofs of the trace-formulas of Grothendieck-Lidskii type for several ideals of nuclear operators). [Comments] LaTeX, English, 17 pp. [Contact e-mail] orein51@mail.ru