Histological image registration is a fundamental task in histological image analysis. It is challenging because of substantial appearance differences due to multiple staining. Keypoint correspondences, i.e., matched keypoint pairs, have been introduced to guide unsupervised deep learning (DL) based registration methods to handle such a registration task. This paper proposes an iterative keypoint correspondence-guided (IKCG) unsupervised network for non-rigid histological image registration. Fixed deep features and learnable deep features are introduced as keypoint descriptors to automatically establish keypoint correspondences, the distance between which is used as a loss function to train the registration network. Fixed deep features extracted from DL networks that are pre-trained on natural image datasets are more discriminative than handcrafted ones, benefiting from the deep and hierarchical nature of DL networks. The intermediate layer outputs of the registration networks trained on histological image datasets are extracted as learnable deep features, which reveal unique information for histological images. An iterative training strategy is adopted to train the registration network and optimize learnable deep features jointly. Benefiting from the excellent matching ability of learnable deep features optimized with the iterative training strategy, the proposed method can solve the local non-rigid large displacement problem, an inevitable problem usually caused by misoperation, such as tears in producing tissue slices. The proposed method is evaluated on the Automatic Non-rigid Histology Image Registration (ANHIR) website and AutomatiC Registration Of Breast cAncer Tissue (ACROBAT) website. It ranked 1st on both websites as of August 6th, 2024.