  We construct B\"acklund transformations (BT) for the Gelfand-Dickey hierarchy
(GD$_n$-hierarchy) on the space of $n$-th order differential operators on the
line. Suppose $L=\partial_x^n-\sum_{i=1}^{n-1}u_i\partial_x^{(i-1)}$ is a
solution of the $j$-th GD$_n$ flow. We prove the following results:
  (1) There exists a system (BT)$_{u,k}$ of non-linear ordinary differential
equations for $h:R^2\to C$ depending on $u_1, \ldots, u_{n-1}$ in $x$ and $t$
variables such that $\tilde L= (\partial+h)^{-1}L(\partial+h)$ is a solution of
the $j$-th GD$_n$ flow if and only if $h$ is a solution of (BT)$_{u,k}$ for
some parameter $k$. Moreover, coefficients of $\tilde L$ are differential
polynomials of $u$ and $h$. We say such $\tilde L$ is obtained from a BT with
parameter $k$ from $L$.
  (2) (BT)$_{u,k}$ is solvable.
  (3) There exists a compatible linear system for $\phi:R^2\to C$ depending on
a parameter $k$, such that if $\phi_1, \ldots, \phi_{n-1}$ are linearly
independent solutions of this linear system then $h:=(\ln W(\phi_1, \ldots,
\phi_{n-1}))_x$ is a solution of (BT)$_{u,k}$ and $(\partial+h)^{-1} L
(\partial+h)$ is a solution of the $j$-th GD$_n$ flow, where
$W(\phi_1,\ldots,\phi_{n-1})$ is the Wronskian Moreover, these give all
solutions of (BT)$_{u,k}$.
  (4) We show that the BT for the GD$_n$ hierarchy constructed by M. Adler is
our BT with parameter $k=0$.
  (5) We construct a permutability formula for our BTs and infinitely many
families of explicit rational solutions and soliton solutions.
