This is an announcement for the paper "Some thoughts on approximation properties" by Oleg Reinov. 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). Archive classification: math.FA Mathematics Subject Classification: 46B28 Spaces of operators, tensor products, approximation Remarks: 17 pages. A talk at "July 22-26 Positivity 2013 Holland, Leiden" Submitted from: orein51@mail.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1403.4746 or http://arXiv.org/abs/1403.4746