Health data exchange is a major challenge due to the sensitive information and the privacy issues entailed. Considering the European context, in which health data must be exchanged between different European Union (EU) Member States, each having a different national regulatory framework as well as different national healthcare structures, the challenge appears even greater. Europe has tried to address this challenge via the epSOS (“Smart Open Services for European Patients”) project in 2008, a European large-scale pilot on cross-border sharing of specific health data and services. The adoption of the framework is an ongoing activity, with most Member States planning its implementation by 2020. Yet, this framework is quite generic and leaves a wide space to each EU Member State regarding the definition of roles, processes, workflows and especially the specific integration with the National Infrastructures for eHealth. The aim of this paper is to present the current landscape of the evolving eHealth infrastructure for cross-border health data exchange in Europe, as a result of past and ongoing initiatives, and illustrate challenges, open issues and limitations through a specific case study describing how Italy is approaching its adoption and accommodates the identified barriers. To this end, the paper discusses ethical, regulatory and organizational issues, also focusing on technical aspects, such as interoperability and cybersecurity. Regarding cybersecurity aspects per se, we present the approach of the KONFIDO EU-funded project, which aims to reinforce trust and security in European cross-border health data exchange by leveraging novel approaches and cutting-edge technologies, such as homomorphic encryption, photonic Physical Unclonable Functions (p-PUF), a Security Information and Event Management (SIEM) system, and blockchain-based auditing. In particular, we explain how KONFIDO will test its outcomes through a dedicated pilot based on a realistic scenario, in which Italy is involved in health data exchange with other European countries.