Cost-effective Na-ion batteries (NIBs) have recently drawn significant interests for their potential applications in grid scale energy storage. NIBs have many advantages over the conventional Li-ion batteries (LIBs), including natural abundance and low costs of Na precursors and significantly reduced fabrication costs arising from Al used as the current collector for anodes. However, developing suitable anode materials for NIBs based on current understanding and experience with LIBs has encountered serious limitations due to the different physicochemical properties of Na from those of Li. This paper critically assesses recent advances in fundamental understanding of Na+ ion storage mechanisms, e.g., phase transition, electrochemical kinetics and morphological evolutions, during the charge/discharge cycles in various anode materials. The current start-of-the-art developments are reviewed of rational design strategies for synthesizing micro/nano-structured anode materials. Challenges and outlooks on further optimization of the structure and the electrochemical performance of NIBs for real-world applications are presented.