The authors present a thorough investigation of (quasi)determinants of matrices over noncommutative rings. The new point of view is that the values of (quasi)determinants are assumed to be contained in the original ring whereas previous authors allowed values in some suitable extension object [cf. \textit{J. Dieudonné}, Bull. Soc. Math. Fr. 71, 27-45 (1943; Zbl 0028.33904), \textit{M. Sato} and \textit{M. Kashiwara}, Proc. Japan. Acad. Sci. 51, 17-19 (1975; Zbl 0337.35067), and \textit{F. A. Berezin}, Introduction to algebra and analysis with anti-commuting variables (1983; Zbl 0527.15020.)] The following basic definition is given. Let \(A=(a_{ij})\) be an \(n\)- square matrix over any unitary ring \(R\), let \(A_{pq}\) denote the \((n- 1)\)-square matrix obtained by deleting the \(p\)-th row and the \(q\)-th column of \(A\), and let \(\xi_{p,q}\) resp. \(\eta_{p,q}\) be the \(p\)-th row vector resp. the \(q\)-th column vector of \(A\) having the entry \(a_{pq}\) omitted. Assuming the existence of \((A_{pq})^{-1}\), the quasideterminant of index \(pq\) of \(A\) is defined by (*) \(| A|_{pq}:=a_{pq}-\xi_{p,q}(A_{pq})^{-1}\eta_{p,q}\), i.e., in general, there are \(n^ 2\) different quasideterminants of a given matrix. If \(R\) happens to be commutative, \((*)\) becomes \(| A|_{pq}=(-1)^{p+q} \text{det} (A)/ \text{det} (A_{pq})\), where det denotes the ``classical'' determinant. Chapter 1 of this paper gives a detailed investigation of several properties of quasideterminants. In Chapter 2, connections between quasiderminants and formal Laurent series as well as representations of bipartite graphs are presented. Chapter 3 deals with representations of ``classical'' determinants, quantum determinants, Berezinians, and Capelli's identity via products of quasideterminants.