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Recent literature on statistical process monitoring pointed out that the quality of products and processes can be often related to profiles, where the function relating a response to one or more location variables (in time or space) is the quality characteristic of interest. An important application of profile monitoring concerns geometric specifications of mechanical components, such as straightness, roundness or free-form tolerance. This paper presents a new approach aimed at extending the method proposed for profile monitoring to surface monitoring. In this case, a geometric specification (such as cylindricity, flatness, etc.) is assumed to characterize the machined surface. The proposed method is based on combining a Spatial Autoregressive Regression (SARX) model (i.e. a regression model with spatial autoregressive errors) to multivariate and univariate control charting. In this work, the approach is applied to a case study concerning surfaces obtained by turning and subject to cylindricity tolerance.