Artificial neural networks have allowed some remarkable progress in fields such as pattern recognition and computer vision. However, the increasing complexity of artificial neural networks presents a challenge for efficient computation. In this thesis, we first introduce a novel matrix multiplication method to reduce the complexity of artificial neural networks, where we demonstrate its suitability to compress fully connected layers of artificial neural networks. Our method outperforms other state-of-the-art methods when tested on standard publicly available datasets. This thesis then focuses on Explainable AI, which can be critical in fields like finance and medicine, as it can provide explanations for some decisions taken by sub-symbolic AI models behaving like a black box such as Artificial neural networks and transformationbased learning approaches. We have also developed a new framework that facilitates the use of Explainable AI with tabular datasets. Our new framework Exmed, enables nonexpert users to prepare data, train models, and apply Explainable AI techniques effectively.Additionally, we propose a new algorithm that identifies the overall influence of input features and minimises the perturbations that alter the decision taken by a given model.Overall, this thesis introduces innovative and comprehensive techniques to enhance the efficiency of fully connected layers in artificial neural networks and provide a new approach to explain their decisions. These methods have significant practical applications in various fields, including portable medical devices. i Another person I would like to thank is Professor Xiangua Xie for all his assistance and support during this Ph.D.