The paper deals with the problem of radar detection of a target echo embedded in Weibull clutter and white Gaussian noise (WGN). Relevant features of the paper, with respect to previous papers on the same subject, refer to the coherent nature of the Weibull process (that modelling the clutter) and of the processing chain. In more detail, the in-phase and quadrature components of the clutter echoes have been modelled to give a Weibull probability density function (PDF) of the amplitude and a uniform PDF of the phase. Any shape of the correlation function among consecutive clutter samples is also allowed in the model. The so called ‘coherent Weibull clutter’ (CWC) introduced in the paper represents a suitable generalisation of the conventional ‘coherent Gaussian clutter’ (CGC). The processing chain is also coherent, i.e. it operates on the in-phase and quadrature components of the signals. To derive a suitable processing scheme, we resort to the general theory of radar detection which applies to any type of PDF and autocorrelation function (ACF) of the target and clutter. Briefly, it is found that the processing scheme is based on two nonlinear estimators of the clutter samples in the two alternative hypotheses (i.e. H0and H1the fully fledged architecture being discussed in the paper. The detection processor turns out to be a suitable generalisation of that concerning the CGC case. A new family of processing schemes is derived in accordance with the statistical model assumed for the useful target. The following cases are covered: target knowna priori, Swerling 0, 1 and 2 models, and partially fluctuating target. Attention is also paid to the interesting case of a target modelled as a coherent Weibull process. The detection of such a target against white Gaussian noise is worked out. The detection performance of the afore-mentioned processing schemes has been evaluated in a number of operational cases of interest. Another novelty of the paper refers to the conception of detection schemes having adaptive features. In more detail, methods are suggested for the on-line estimation of the parameters of the two nonlinear estimators of the clutter. Results are presented concerning the detection loss against the number of range cells along which the average of parameters estimate is performed. Another adaptive feature explored refers to the on-line setting of a constant false-alarm rate (CFAR) detection threshold. Also in this case, an evaluation of the corresponding detection loss is made. To summarise, the paper represents a suitable generalisation of the detection theory of a coherent Gaussian target embedded in a CGC. This new theory proves its usefulness in advanced radar systems that should have high performance, notwithstanding challenging environments including spiky clutter and/or spiky target (e.g. stealth).