This paper introduces and motivates the use of Gaussian Mixture Models (GMMs) for on-line signature verification. The individual Gaussian components are shown to represent some local, signer-dependent features that characterise spatial and temporal aspects of a signature, and are effective for modelling its specificity. The focus of this work is on automated order selection for signature models, based on the Minimum Description Length (MDL) principle. A complete experimental evaluation of the Gaussian Mixture signature models is conducted on a 50-user subset of the MCYT multimodal database. Algorithmic issues are explored and comparisons to other commonly used on-line signature modelling techniques based on Hidden Markov Models (HMMs) are made.