Vulcan

Vulcan is a hypothetical planet that was thought to orbit the Sun in an orbit inner to Mercury before the XX Century, named after the Roman god equivalent to the Greek

Theory and observations
Claims about objects seen transiting in front of the Sun have existed since the invention of the telescope, back in the XVII Century and reaching a maximum in the XIX Century, even if most if not all of them are more than likely to have been just reports of the observations of sunspots if real at all.

Theories about an innermost planet to Mercury began circulating in such century and were followed by observations that hoped to catch such planet when it passed in front of the Sun, given that seeing it after sunset or before dawn would be very difficult or outright impossible getting lost in the Sun's glare. All of them, of course, came empty.

Enter Einstein
Anomalies in Mercury's orbit that models of the latter failed to account for, causing astronomers to think the gravity of planet(s) orbiting closer to the Sun were responsible, were a mystery until 1915, when Einstein's Theory of Relativity showed they were caused by space-time being warped by the Sun, matching exactly the observations done in the previous century, with such prediction being confirmed during the solar eclipse of May 29, 1919 catapulting Einstein to fame and Vulcan's existence being no longer needed.

Today
Earth-based searches have been supplemented by observations from space using from telescopes as to spacecrafts that have boldly gone where no spacecraft has gone before sent within Mercury's orbit as NASA's  that not only have failed to find such planet but also that the space within Mercury's orbit is devoid of bodies of significant size with a  that would be there having also failed to materialize, besides the occasional comet or It should also be noted that many other planetary systems contain planets  within a radius equivalent to Mercury's orbit. Wether this is simply an observational bias, as such kind of systems are easier to detect with our present technology, or the lack of them in our Solar System is a feature and not a bug is a matter of research.