Designing advanced material systems poses challenges in integrating knowledge and representation from multiple disciplines and domains such as materials, manufacturing, structural mechanics, and design optimization. Data-driven machine learning and computational design methods provide a seamless integration of predictive materials modeling, manufacturing, and design optimization, enabling the accelerated design and deployment of advanced material systems. In this talk, we will introduce state-of-the-art data-driven methods for designing heterogeneous nano- and microstructural materials and complex multiscale metamaterial systems. We will discuss research developments in design representation, design evaluation, and design synthesis, along with novel design methods that integrate machine learning, mixed-variable Gaussian process modeling, Bayesian optimization, topology optimization, and the concept of digital twins. Furthermore, we will address the challenges and opportunities involved in designing engineered material systems.