Let \(T^d\) be the \(d\)-dimensional torus. In the space \(L_q:= L_q (T^d)\) the author investigates two kinds of widths of the unit ball \(SB^r_{p, \theta}\) of the mixed smoothness Besov space and of the unit ball \(SW^r_p\) of the mixed smoothness Sobolev space: on the one hand the well-known entropy number \(\varepsilon_n\), on the other hand the width \(\rho_n\) recently introduced by \textit{V. Maiorov} and \textit{J. Ratsaby} [Constructive Approximation 15, No. 2, 291-300 (1999; Zbl 0954.41016)]. For \(10\) there are found upper bounds for \(\varepsilon_n (SB^r_{p, \theta}, L_q)\) and \(\varepsilon_n (SW^r_p,L_q)\) as well as lower bounds for \(\rho_n (SB^r_{p, \theta}, L_q)\) and \(\rho_n(SW^r_p,L_q)\). Taking into consideration these estimates there are established asymptotic orders of the investigated widths.