5G and B5G/6G foundations heavily rely on virtualization technologies, with virtualized Radio Access Networks (vRANs) being a major cornerstone. While vRANs have traditionally suffered from significant hardware/software coupling, next-generation vRANs aim for open, standardized interfaces and multi-vendor, interoperable components to enable truly flexible deployments following cloud-native principles. The O-RAN Alliance is promoting a novel Open RAN architecture to further boost flexibility and cost efficiency. To reduce costs and effectively achieve promised disaggregation levels, O-RAN must ensure shared, integrated transport networks rather than dedicated, overprovisioned links characteristic of traditional approaches. However, maintaining deterministic performance requirements in such cost-effective networks (i.e., general-purpose Ethernet networks), especially in time-critical interfaces, presents a challenge. This article reviews the most relevant Time Sensitive Networking (TSN) standards that can significantly benefit O-RAN, including IEEE 802.1CM, IEEE 802.1Qbu, and IEEE 802.1Qbv, for providing determinism over cost-efficient networks. The paper explores the design space for a TSN-enabled O-RAN architecture, reporting on its requirements and deployment options. Finally, it discusses the opportunities and challenges O-RAN will face when adopting TSN technologies to fully open the vRAN ecosystem.