We discuss issues related to the use of the normality assumption in statistical process monitoring with continuous data. Our illustrations involve the Shewhart X-chart. We illustrate some of the dangers and pitfalls in using nonlinear transformations to obtain the approximate normality of the process data. We argue that such transformations should rarely be made and never made before assessing and establishing process stability through control charts and, if necessary, process improvements. The nonlinear transformation process can mask outliers, the importance of which needs to be assessed by the process engineers or other domain experts. For clearly non-normal in-control processes, we recommend the use of an appropriate fitted distribution to obtain control limits in the ongoing monitoring of Phase II or the use of nonparametric control charts. We show that the low power of goodness-of-fit normality tests can lead to an unexpectedly poor in-control statistical performance in Phase II when the normality assumption is made incorrectly.