The paper is devoted to investigation of ``small'' operators on symmetric spaces over von Neumann algebras and Köthe-Bochner spaces over finite measure spaces. Let ``operator'' mean linear bounded operator. Theorem~3.8 asserts that, under mild assumptions on a symmetric space \(E\) on \([0,1]\) and a semifinite von Neumann algebra \(\mathcal{M}\), an operator \(T\) from \(E(\mathcal{M},\tau)\) to a Banach space \(X\) is strongly Dunford-Pettis if and only if \(T \circ i \colon M \cap E(\mathcal{M},\tau) \to_i E(\mathcal{M},\tau) \to_T X\) is compact. Theorem~5.2 concerns narrow operators: if \(1 \le p,r < 2\) then every operator \(T \colon L_p \to C_r\) is narrow, where \(C_r\) is a certain Schatten-von Neumann ideal of compact operators (the nontrivial contribution concerns the case where \(1 < p < 2\) and \(1 \le r \le p\), which was an open problem). The following two results concern operators \(T \colon L_1(\nu,X) \to c_0\), where \((A, \Sigma, \nu)\) is a finite measure space and \(X\) a reflexive Banach space. According to Theorem~4.9, \(T\) is Dunford-Pettis if and only if \(T\) is dominated. Theorem~5.8 asserts that, if moreover, \(\nu\) is atomless then every Dunford-Pettis operator \(T \colon L_1(\nu,X) \to c_0\) is narrow.