A variety of $n$-groupoids (i.e. algebras with one $n$-ary operation $f$) is said to be a primitive $n$-variety if it is defined by a system of identities of the following form: $$ f(x_{i_1},x_{i_2},łdots,x_{i_n}) = f(x_{j_1},x_{j_2},łdots,x_{j_n}) $$ Here we give a convenient description of free objects in primitive $n$-varieties, and several properties of free objects are also established.