Bipartite graphs are a special type of graph data structure where vertices can be divided into two disjoint and independent sets, and each edge connects a vertex from one set to a vertex in the other set. They can be used to model many real-world applications such as user-item interaction networks, authorship networks, and product-customer networks. Despite their growing importance and popularity in recent years, bipartite graphs remain relatively unexplored compared to unipartite graphs. This thesis focuses on three unexplored problems on bipartite graphs: sign prediction, graph classification, and follower prediction. Using increasingly powerful artificial neural network techniques, we have developed effective graph neural networks tailored to address these three problems. Firstly, we focus on the sign prediction problem on signed bipartite graphs. Signed bipartite graphs can represent relationships such as buyers reviewing products, with positive and negative signs on the edges. To predict the signs of edges, in this thesis, we propose the first graph neural network on signed bipartite graphs, namely the Polarity-based Graph Convolutional Network (PbGCN), with the help of balance theory. We introduce a novel polarity attribute to the signed bipartite graph, based on which we construct a one-mode projection to establish connections between same-type nodes with similar polarity values. This allows PbGCN to directly aggregate information between same-type nodes, overcoming the limitations of prior work and achieving strong performance. Second, we study the graph classification problem on bipartite graphs. Bipartite graph classification is an important tool and can be used to solve many real-world problems such as classifying users into different language or topic groups in user-wikipage editing bipartite graphs, anti-money laundering in e-commerce platforms. Although several graph classification methods have been proposed for unipartite and homogeneous graphs using kernel methods and graph neural networks, these methods are unable to effectively capture the hidden information in bipartite graphs. In this thesis, we propose the first bipartite graph-based capsule network, namely Bipartite Capsule Graph Neural Network (BCGNN), for the bipartite graph classification task. BCGNN leverages the capsule network and obtains information between the same types of vertices in the bipartite graphs by constructing the one-mode projection. Third, we study the follower prediction problem on temporal bipartite graphs. The follower prediction problem is to predict the likelihood that two visitors (i.e., a follower and a leader) will visit the same location within a specific time window. This problem is important for preventing the widespread transmission of infectious diseases in communities. In this thesis, we propose a model called Follower Prediction Graph Network (FPGN) to solve this problem. FPGN is inspired by the state-of-the-art temporal graph edge prediction algorithm TGN and addresses the shortcomings of existing algorithms. It utilizes graph structure information, time interval statistics, and a GAT-based prediction module to achieve high accuracy in follower prediction. Finally, we conducted extensive empirical studies of the proposed techniques on real- world datasets. The results demonstrate the effectiveness and superiority of our proposed methods.