Let x be an element of a group G and be an automorphism of G. Then for a positive integer n, the autocommutator $[x,_n{alpha}]$ is defined inductively by $[x,{alpha}]=x^{-1}x^{alpha}=x^{-1}{alpha}(x)$ and $[x,_{n+1}{alpha}]=[[x,_n{alpha}],{alpha}]$. We call the group G to be n-auto-Engel if $[x,_n{alpha}]=[{alpha},_nx]=1$ for all $x{in}G$ and every ${alpha}{in}Aut(G)$, where $[{alpha},x]=[x,{alpha}]^{-1}$. Also, for any integer $n{ eq}0$, 1, a group G is called an n-auto-Bell group when $[x^n,{alpha}]=[x,{alpha}^n]$ for every $x{in}G$ and each ${alpha}{in}Aut(G)$. In this paper, we investigate the properties of such groups and show that if G is an n-auto-Bell group, then the factor group $G/L_3(G)$ has finite exponent dividing 2n(n-1), where $L_3(G)$ is the third term of the upper autocentral series of G. Also, we give some examples and results about n-auto-Bell abelian groups.