Recently, food safety and quality have become critical issues of great concern throughout the world. Fish is one of the most vulnerable and perishable aquatic products. The evaluation of fish and fillet freshness is therefore very significant in research and development for providing premium and supreme quality for human health and acceptance by consumers, as well as for international trade. The texture and structure of fish muscle are important freshness quality attributes that depend on several parameters such as hardness, cohesiveness, springiness, chewiness, resilience, and adhesiveness, as well as the internal cross-linking of connective tissue and the detachment of fibers. This review aims to present recent advances of texture and structure measurements and analyses, including sensory evaluation and instrumental methods, for indicating and evaluating fish freshness quality. Factors affecting these measurements are detailed and correlations between texture and structure are discussed. Moreover, the limitations and challenges of fish texture and structure measurements are described and some viewpoints about current work and future trends are also presented.