The paper describes an analysis of adhesion at the contact between non-Gaussian rough surfaces using the Weibull distribution with skewness as the key parameter to characterize asymmetry. The analysis uses an improved elastic–plastic model of contact deformation that is based on accurate Finite Element Analysis (FEA) of an elastic–plastic single asperity contact. Large range of interference values is considered starting from fully elastic through elastic–plastic to fully plastic regime of contacting asperities. The well-established elastic and plastic adhesion indices are used to consider the different conditions that arise as a result of varying load and material parameters. The loading and unloading behaviour for different combinations of the adhesion indices and skewness values are obtained as functions of mean separation between the surfaces. Transitional values of adhesion indices and skewness at which the influence of surface forces becomes insignificant are found to depend on material and surface parameters. Comparison with studies using previous elastic–plastic model that was based on some arbitrary assumptions shows significant differences in loading behaviour.