In the last two decades a considerable number of results concerning various kinds of boundedness in topological groups appeared in the literature: $\mathsf M$-boundedness, $\mathsf H$-boundedness, $\mathsf R$-boundedness, $\mathsf S$-boundedness. These classes of groups have been defined by Kočinac in 1998, and the class of $\mathsf M$-bounded groups was independently introduced by Okunev and Tkachenko in 1998 under the name $\mathsf o$-bounded groups. For more information about these classes see, for example, [3,4,9--11,16,19,23]. In this paper we introduce in a similar way several properties in the class of topologized groups. Our study is mainly related to paratopological groups and differences between these properties in topological and paratopological groups. For more details on paratopological groups see [2,20--22] and also the papers [1,5,8,13--15,17,18].