The Discourse on Comets (Italian: Discorso delle Comete) was a pamphlet published in 1619 with Mario Guiducci as the named author, though in reality it was mostly the work of Galileo Galilei. In it Galileo conjectured that comets were not physical bodies but atmospheric effects like the aurora borealis.

Three comets were seen in Europe in 1618. The first appeared in October, the second in mid-November, and the third and brightest at the end of November. After publishing Letters on Sunspots in 1613, Galileo had largely stopped working on telescopic astronomy and published nothing further based on observing and recording astronomical events. In 1616 the heliocentric views of Copernicus were declared formally heretical and Galileo was warned by Cardinal Bellarmine to neither teach nor defend them. After this he was silent on astronomical matters for several years. However Virginio Cesarini wrote to him asking for his views on the 1618 comets, as did Archduke Leopold of Austria and Domenico Bonsi, who wrote to him that the court mathematicians of Louis XIII of France all wanted to know his opinion on the phenomenon.

Galileo had not observed the comets as he was unwell in the autumn of 1618. However he learned that the Collegio Romano had held four lectures on the comets, respectively by a theologian, a mathematician, a philosopher and a rhetorician. The mathematician was Orazio Grassi, a pupil of Maelcote and Grienberger. Grassi argued that the absence of parallax meant that the comets must be very distant from the Earth, and he suggested that they existed beyond the Moon. Soon afterwards the lecture was published in Rome as an anonymous pamphlet entitled "De Tribus Cometis Anni MDCXVIII".

In March 1619, Galileo received a letter from Giovanni Battista Rinuccini alerting him to the fact that some people outside the Jesuit order were claiming that Grassi's lecture on comets provided a definitive proof that Copernicus' ideas were wrong. The evident threat to Copernicus, whom Galileo could no longer defend, prompted him to attack the Tychonic ideas now popular among the Jesuits with particular force.

Galileo received a copy of Grassi's lecture and was very angered by it. The notes he scribbled in the margin of his copy are full of insults - 'pezzo d'asinaccio' ('piece of utter stupidity'), 'bufolaccio' ('buffoon'), 'villan poltrone' ('wicked idiot'), 'balordone' ('bumbling idiot'). He decided to respond to it through his friend Mario Guiducci, who wanted a topic for his planned address to the Accademia Fiorentina. Guiducci read the Discourse on Comets at the Accademia Fiorentina in May 1619 and it was published the next month.

In public, Galileo insisted that Guiducci, and not he, was the author of the Discourse on Comets. Despite Galileo's public protestations, there is no doubt whatever that he was the main author of the Discourse on Comets. The manuscript is largely in Galileo's handwriting, and the sections in Guiducci's hand have been revised and corrected by Galileo.

Grassi responded to the Discourse on Comets as if it were propounding a theory about their origin, but Galileo made clear that he was not doing that. Rather, as in his Letter to the Grand Duchess Christina four years previously, he was insisting that the burden of proof lay with those who had ideas that did not accord with his own. He intended to criticise pedantic thinkers who believed they had easily found a definitive answer to something, disregarding the fact that nature may have many possible ways of producing the same effect. He conceded that he knew very little about comets. His point was to expose those who were convinced that they knew the answers. The Discourse does not attempt to offer clear proofs of Galileo's conjectures, (unlike his Letters on Sunspots or Discourse on Floating Bodies); instead it focuses on arguments that undermine Grassi's contentions, forcing him to examine the phenomenon of comets more thoroughly and produce more substantial evidence for his argument that they are real.

The Discourse on Comets, although formally a response to Grassi, was a rebuttal of the arguments made by Tycho Brahe. It advanced the proposition that the absence of parallax observable with comets was due not to their great distance from the Earth, but to the fact that they were not real objects; they were probably atmospheric effects. Galileo (through Guiducci) also argued against Tycho's case for comets having uniform, circular paths. Instead, he maintained, their paths were straight. Throughout, he expressed surprise that mathematicians at the Collegio (where Guiducci had been educated) could have adopted Tycho's positions so uncritically when his arguments were so poor. As well as attacking Grassi, the Discourse also continued an earlier dispute with Christoph Scheiner about sunspots, belittling the illustrations in Scheiner's book as 'badly coloured and poorly drawn.'