Robust methods for precise segmentation of breast region or volume from breast X-ray images, including mammogram and tomosynthetic image, is crucial for applications of these medical images. However, this task is challenging because the acquired images not only are inherent noisy and inhomogeneous, but there are also connected or overlapped artifacts, or noises on the images as well, due to local volume effect of tissues, parametric resolutions and other physical limitations of the imaging device. This paper proposes and develops robust fuzzy c-means (FCM) segmentation methods for segmentation of breast region on breast x-ray images, including mammography and tomosynthesis, respectively. We develop spatial information- and kernel function- based FCM methods to differentiate breast area or breast volume. Spatial information based FCM method incorporates neighborhood pixels' intensities into segmentation because neighbored pixels on an image are highly correlated. Kernel based FCM algorithm is developed by transforming pixel intensity using kernel functions to better improve segmentation performance. The proposed segmentation methods are implemented on mammograms and tomosynthetic images and compared with conventional FCM results. Experiment results demonstrate the proposed segmentation methods are much better compared with traditional FCM method, and are more robust to noises. The developed kernel and spatial based FCM method will be applied for differentiation of breast density and abnormal regions within the breast region to examine its performance in reducing false positive segmentations.