For more than 18 years, the Estonian electronic identity card (ID card) has provided a secure electronic identity for Estonian residents. The public-key cryptography and private keys stored on the card enable Estonian ID card holders to access e-services, give legally binding digital signatures and even cast an i-vote in national elections. This work provides a comprehensive study on the Estonian ID card and its security challenges. We introduce the Estonian ID card and its ecosystem by describing the involved parties and processes, the core electronic functionality of the ID card, related technical and legal concepts, and the related issues. We describe the ID card smart card chip platforms used over the years and the identity document types that have been issued using these platforms. We present a detailed analysis of the asymmetric cryptography functionality provided by each ID card platform and present a description and security analysis of the ID card remote update solutions that have been provided for each ID card platform. As yet another contribution of this work, we present a systematic study of security incidents and similar issues the Estonian ID card has experienced over the years. We describe the technical nature of the issue, mitigation measures applied and the reflections on the media. In the course of this research, several previously unknown security issues were discovered and reported to the involved parties. The research has been based on publicly available documentation, collection of ID card certificates in circulation, information reflected in media, information from the involved parties, and our own analysis and experiments performed in the field.