Since primeval times, the appearance of bright comets has impressed humanity. But more detailed conclusions about individual comets or even their paths can only be obtained from written records, which are available from about 500 BC. The most complete reports of comets come from China, whereas reports from Europe are incomplete, and only single out certain individual comets. Chinese astronomers also gave details of the oldest cometary apparition that can be identified nowadays, from the year 613 BC. Readily understood indicators are provided by Halley's Comet, which was observed in the years 240, 164, 87 and 12 BC. After the birth of Christ, it appeared in AD 66, 141, 218, 295, 374 and 451.

One of the oldest comets documented from Europe is the Great Comet observed by Aristotle in the winter of 373–372 BC. The Greek thinker saw it in his childhood. He later wrote in his work Meteorologica: “its light stretched as a great ribbon across one third of the sky”. Aristotle linked the comet with an earthquake and with a ‘tidal wave’ that the latter produced – an early example of the interpretation of comets as harbingers of disaster that persisted into modern times. According to Diodorus, witnesses claimed that the comet had cast shadows like the Moon. A few chroniclers report that the comet eventually ‘broke into two planets’. According to modern analysis, it was probably a sungrazer, which broke up after a close approach to the Sun. If this is so, the Comet of 372 BC may be the parent comet of the Great Comets of 1680 and 1843.

Dating from the year 240 BC, the first full sightings of Comet Halley come from China and Mesopotamia. At its return in 164 BC the apparition of the comet was documented on Babylonian cuneiform tablets. The comet of 135 BC is famous for appearing at the birth of King Mithridates VI of Pontus. It was visible for 70 days; Seneca described it, according to eye-witnesses, as “stretching out like the Milky Way”.

Ettore Majorana contributed several ideas that have had a significant, lasting impact on condensed matter physics, broadly construed. In this chapter I will discuss, from a modern perspective, four important topics that have deep roots in Majorana's work.

1. Spin response and universal connection: In paper N.6, Majorana considered the coupling of spins to magnetic fields. The paper is brief, but it contains two ingenious ideas whose importance extends well beyond the immediate problem he treated, and should be part of every physicist's toolkit. The first of those ideas is that, having solved the problem for spin 1/2, one can deduce the solution for general spins by pure algebra. Majorana's original construction uses a rather specialized mathematical apparatus. Bloch and Rabi, in a classic paper [62], brought it close to the form discussed in Section 14.1. Rephrased in modern terms, it is a realization – the first, I think, in physics – of the universality of non-Abelian charge transport (Wilson lines).

2. Level crossing and generalized Laplace transform: In the same paper, Majorana used an elegant mathematical technique to solve the hard part of the spin dynamics, which occurs near level crossings. This technique involves, at its center, a more general version of the Laplace transform than its usual use in constant-coefficient differential equations. Among other things, it gives an independent and transparent derivation of the celebrated Landau–Zener formula for non-adiabatic transitions. (Historically, Majorana's work on the problem, and also that of Stückelberg [75], was essentially simultaneous with Landau's [73] and Zener's [74].) Majorana's method is smooth and capable of considerable generalization. It continues to be relevant to contemporary problems.

3. Majorana fermions: from neutrinos to electrons: Majorana's most famous paper N.9 concerns the possibility of formulating a purely real version of the Dirac equation. In a modern interpretation, this is the problem of formulating equations for the description of spin-1/2 particles that are their own antiparticles: Majorana fermions. Majorana's original investigation was stimulated in part by issues of mathematical esthetics, and in part by the physical problem of describing the then hypothetical neutrino.