Solutions to the exact model-matching and block-decoupling problems for both continuous- and discrete-time linear time-varying systems are presented. The parametrisation of the whole class of proper solutions is given. For the decoupling, the minimal delay problem is also considered in a time-varying setting. The approach is algebraic and based on the Smith-MacMillan form at infinity of a transfer matrix of a time-varying system which has been recently introduced in systems theory. This avoids the difficulties related to the inversion of the transfer matrices with entries in non-commutative fields over which the determinants (of Dieudonne or Ore type) are much more complicated. The solutions presented here involve only standard matrix computations excluding direct matrix inversions and are thus easy to implement in practice. Examples are treated in detail to illustrate the theoretical results and the way in which the computations are done and a physical example is also shown.