The Internet of Things (IoT) is rapidly becoming an integral component of the industrial market in areas such as automation and analytics, giving rise to what is termed as the Industrial IoT (IIoT). The IIoT promises innovative business models in various industrial domains by providing ubiquitous connectivity, efficient data analytics tools, and better decision support systems for a better market competitiveness. However, IIoT deployments are vulnerable to a variety of security threats at various levels of the connectivity and communications infrastructure. The complex nature of the IIoT infrastructure means that availability, confidentiality and integrity are difficult to guarantee, leading to a potential distrust in the network operations and concerns of loss of critical infrastructure, compromised safety of network end-users and privacy breaches on sensitive information. This work attempts to look at the requirements currently specified for a secure IIoT ecosystem in industry standards, such as Industrial Internet Consortium (IIC) and OpenFog Consortium, and to what extent current IIoT connectivity protocols and platforms hold up to the standards with regard to security and privacy. The paper also discusses possible future research directions to enhance the security, privacy and safety of the IIoT.