A tale of analogies: gravitomagnetic effects, rotating sources, observers and all that

Gravitoelectromagnetic analogies are somewhat ubiquitous in General Relativity, and they are often used to explain peculiar effects of Einstein's theory of gravity in terms of familiar results from classical electromagnetism. Perhaps, the best known of these analogy pertains to the similarity between the equations of electromagnetism and those of the linearized theory of General Relativity. But the analogy is somewhat deeper and ultimately rooted in the splitting of spacetime, which is preliminary to the definition of the measurement process. In this paper we review the various approaches that lead to the introduction of a magnetic-like part of the gravitational interaction, briefly called gravitomagnetic and, then, we provide a survey of the recent developments both from the theoretical and experimental viewpoints.