In \S1 we construct the formal theory of quasideterminants, including an important Sylvester identity. In \S2 we study the theory of noncommutative continued fractions. In \S3 we provide expansions of quasideterminants into formal power series connected with noncommutative characteristic functions of graphs. Applications are in \S4. We explain how ordinary and quantum determinants and the Capelli identity can be deduced from our quasideterminants. We also provide the definition of Bäcklund transformations in our theory and make a few initial steps towards the theory of noncommutative representations. This theory differs significantly from the commutative one. For example, there exist natural ``Plücker coordinates'' of flag manifolds, while the notion of the Grassmannian is apparently meaningless. In the last subsection we give a solution for noncommutative Toda lattices.