  Renz (1996), Ouchti(2005), El Machkouri and Ouchti (2007) and Mourrat (2013)
have established the bounds on the rate of convergence in the central limit
theorem for discrete time martingales. In the present paper a modification of
the methods, developed by Bolthausen (1982) and Grama and Haeusler (2000), is
applied for obtaining exact rates of convergence in the central limit theorem
for martingales with differences having conditional moments of order $2+\rho,
\rho>0$. Our results significantly improve and generalise the bounds of Renz
(1996), Ouchti(2005), El Machkouri and Ouchti (2007) and Mourrat (2013). Our
results generalise and strengthen the bounds mentioned above. An application to
Lipschitz functionals of independent random variables is also given.
