Digital Twin (DT) systems are revolutionizing modern industry by enabling real-time monitoring, simulation, and predictive control of physical assets. However, their widespread adoption in critical domains is contingent upon the trust and security they inspire. This paper presents a comprehensive survey of trust and security in DT systems, synthesizing recent advancements to bridge interdisciplinary gaps. We propose a novel taxonomy that categorizes trust into behavioral and non-behavioral dimensions and aligns these with the architectural layers of a DT. The survey meticulously analyzes the evolving threat landscape, detailing DT-specific vulnerabilities and their implications across diverse application domains. Furthermore, we explore current defense mechanisms, architectural models for secure data distribution, and privacy-preserving techniques such as federated learning and differential privacy. The paper also investigates trust-building strategies, including certification, explainable AI, and stakeholder-centric design. Finally, we identify critical open challenges and outline promising future research directions, including the need for unified trust metrics, lightweight security for edge DTs, and resilient, adaptive autonomy. This survey serves as a foundational reference for researchers and practitioners aiming to develop intelligent, connected, and inherently trustworthy digital twin ecosystems.