Certain classes of continuous linear operators in Banach and locally convex spaces are studied. A characterization of operators $T:X\to Y$, transforming bounded sets of the Banach space $X$ into conditionally weakly compact sets of the Banach space $Y$, is given, and also a particular case where $X=C(K)$ is considered. It is proved that if $E$ is a Fréchet space and $F$ is a complete ($\mathscr{DF}$)-space, then the classes of absolutely summing and Nikodýmizing operators from $E$ into $F$ coincide.