From 2000 to 2010, graph Laplacians had been widely used in dimensional reduction methods, clustering methods, and semi-supervised learning (i.e., Laplacian Eigenmaps, spectral clustering, and graph-based semi-supervised learning). The applications of these methods are huge such as image retrieval (Laplacian Eigenmaps), speech separation (spectral clustering), and protein function prediction (graph Laplacian based semi-supervised learning method). In this thesis, the dimensional reduction methods, clustering methods, and classification methods for hypergraph data structure (i.e., utilizing the hypergraph Laplacian) will be developed. This work includes the classic machine learning methods and modern deep learning methods for hypergraph data structure.In specific, first, the definitions of the three hypergraph Laplacians will be represented clearly. Next, the novel weighted un-normalized hypergraph Laplacian Eigenmaps and the weighted un-normalized hypergraph Laplacian based semi-supervised learning method will be developed and represented clearly. Last but not least, these proposed algorithms will be tested on the zoo dataset available from UCI repository and the tiny version of the 20 newsgroups dataset.Second, the definitions of the gradient and divergence operators of hypergraph will be introduced. Next, the definition of Laplace operator of hypergraph and its properties will be represented. Then, the definition of the curvature operator of hypergraph and its properties will be shown clearly. Next, the definition of the p-Laplace operator of hypergraph and its properties will be presented. Then, we will show how to derive the algorithm of the un-normalized hypergraph p-Laplacian based semi-supervised learning method from the regularization framework. Finally, the accuracy performance measures of the un-normalized hypergraph Laplacian based semi-supervised learning algorithm (i.e., the current state of art method) will be compared with the un-normalized hypergraph p-Laplacian based semi-supervised learning algorithms (i.e., our proposed methods).Third, the clustering problem will be defined and the novel graph/hypergraph convolutional neural network-based clustering technique will be represented clearly. Next, the two Citeseer dataset and the Cora dataset that will be used in this part will be described. Finally, the performance of the hypergraph convolutional neural network based-clustering technique (i.e., our proposed method) will be compared with the performances of the graph convolutional neural network based-clustering technique, the k-means clustering technique and the spectral clustering technique testing on these two Citeseer and Cora datasets.Fourth, the neural network/deep learning methods for hypergraph data structure are developed and represented clearly and will be utilized to solve the classification task. In details, these methods are utilized to solve the image classification task.Finally, the neural network/deep learning methods for directed hypergraph data structure are developed and represented clearly and will be utilized to solve the classification task.