In this paper we construct (quasi) determinants of matrices over noncommutative rings. Unlike the well-known approach of J. Dieudonne's and his followers, we consider (quasi)-determinants with values in the initial ring and not in an associated object. The construction and the results of our paper has a category character, ie, can be carried over wordfor-word to matrices of morphisms of additive categories. We give formulas which express determinants of various kinds (the determinant of a matrix over a commutative ring, the Berezinian, the quantum determinant, the determinants occurring in Capelli's identity) in terms of our quasideterminants. All these formulas have similar expressions. For the description of quasideterminants we make use of the representation of bipartite graphs.