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 is an element of(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 GrothendieckLidskii type for several ideals of nuclear operators).