Since it is an unsupervised data analysis approach, clustering relies solely on the location of the data points in the data space or, alternatively, on their relative distances or similarities. As a consequence, clustering can suffer from the presence of noisy data points and outliers, which can obscure the structure of the clusters in the data and thus may drive clustering algorithms to yield suboptimal or even misleading results. Fuzzy clustering is no exception in this respect, although it features an aspect of robustness, due to which outliers and generally data points that are atypical for the clusters in the data have a lesser influence on the cluster parameters. Starting from this aspect, we provide in this paper an overview of different approaches with which fuzzy clustering can be made less sensitive to noise and outliers and categorize them according to the component of standard fuzzy clustering they modify.