The aim of this paper is to present the notions of actions and coverings of $\text{cat}^1$-groupoids and to prove the natural equivalence between their categories. Moreover, in this context, we characterize the quotient concept of $\text{cat}^1$-groupoids. Finally we extend these notions to $\text{cat}^n$-groupoids which are groupoid version of $\text{cat}^n$-groups.