The paper considers spaces of periodic functions of several variables, namely, the Lorentz space Lq,τ(Tm), the class of functions with bounded mixed fractional derivative Wr¯¯¯q,τ, 1<q,τ<∞, and studies the order of the best M-term approximation of a function f∈Lp,τ(Tm) by trigonometric polynomials. The article consists of the introduction, the main part, and the conclusion. In the introduction, we introduce basic concepts, definitions, and necessary statements for the proof of the main results. You can also find information about previous results on the topic. In the main part, we establish exact-order estimates for the best M-term approximations of functions of the class Wr¯¯¯q,τ1 in the norm of the space Lp,τ2(Tm) for various relations between the parameters p,q,τ1,τ2.</q,τ<∞