  Rosser theories play an important role in the study of the incompleteness
phenomenon and mete-mathematics of arithmetic. In this paper, we first define
notions of $n$-Rosser theories, exact $n$-Rosser theories, effectively
$n$-Rosser theories and effectively exact $n$-Rosser theories (see Definition
1.6). Our definitions are not restricted to arithmetic languages. Then we
systematically examine properties of $n$-Rosser theories and relationships
among them. Especially, we generalize some important theorems about Rosser
theories for RE sets in the literature to $n$-Rosser theories in a general
setting.
