The visual world composed by the human and computational cognitive systems strongly relies on shapes of objects. Skeleton is a widely applied shape feature that plays an important role in many fields of image processing, pattern recognition, and computer vision. Thinning is a frequently used, iterative object reduction strategy for skeletonization. Sequential thinning algorithms, which are based on contour tracking, delete just one border point at a time. Most of them have the disadvantage of order-dependence, i.e., for dissimilar visiting orders of object points, they may generate different skeletons. In this work, we give a survey of our results on order-independent thinning: we introduce some sequential algorithms that produce identical skeletons for any visiting orders, and we also present some sufficient conditions for the order-independence of template-based sequential algorithms.