The coupling of surface shape dynamics and surface composition for a material growing by electrodeposition has been found in recent modelling work by the authors, to give rise to a rich morphogenetic scenario. In this paper we concentrate on a systematic description of morphogenesis occurring during metal and alloy electrodeposition. First of all we give a brief presentation of the physical facts relevant to the generation of shapes in electrochemical growth processes, essentially the balance between change-transfer and mass-transport rates. Hence, we review the mathematical modelling work recently published by the authors, based on analytical and numerical studies of a system of reaction-diffusion equations for morphological and adsorption dynamics. Eventually, a selection of examples of experimental validation of the model by means of numerical simulations is proposed.