We explore the fundamental properties that are necessary to ensure that election schemes behave as expected. The exploration reveals how our understanding of those expectations has evolved, culminating in the emergence of formal definitions of properties necessary to fulfil expectations. We provide insights into definitions of secrecy and verifiability, allowing us to learn and appreciate the underlying intuition and technical details of these notions. Equipped with definitions, we can build election schemes that can be proven to behave as expected. And, as an illustrative example, we review a variant of the Helios election system that was built and proven secure, in this way. Furthermore, the definitions can be used to analyse existing election schemes, and vulnerabilities have been uncovered. Indeed, we describe a series of vulnerabilities that were discovered during the analysis of the original Helios system, which advanced our understanding of system behaviour and prompted the design of the aforementioned variant. Thus, this article contributes to the science of security by sharing valuable insights into elections, and demonstrating the value that formal definitions and analysis have in building schemes guaranteed to behave as expected.