The problem of constructing adaptive minimum bit error rate (MBER) decision feedback equalisers (DFE's) for binary signalling is considered. Gradient and Gauss-Newton algorithms are considered for both conventional and state (or space) translation forms of the DFE. The Hessian matrix for the Gauss-Newton algorithm is introduced for the rst time. Kernel density estimation is demonstrated to provide a convenient mechanism for approximating the BER as a smooth function of the available data. This leads to the development of a number of block and serial adaptive algorithms. Computer simulation is used to assess the performance of these algorithms.