The mathematics
Archimedes' surviving works are the summit of Greek mathematics: on the sphere and cylinder, on floating bodies, on the equilibrium of planes, on spirals, and the measurement of the circle. He obtained results — including the volume and surface of a sphere, and a rigorous bound on pi — by methods of exhaustion that anticipate integration by roughly nineteen centuries.
The Method, in which he explains how he found results before proving them, was recovered in 1906 from a palimpsest whose parchment had been scraped and reused for a prayer book. It was studied again with modern imaging from the 1990s, and it shows him reasoning mechanically to reach conclusions he then proved geometrically.
The engineering
He is remembered popularly for machines: the compound pulley, the water screw associated with his name, and the devices said to have defended Syracuse against the Roman siege of 213–212 BCE — cranes that lifted ships out of the water, and catapults ranged for different distances.
Plutarch's account is the fullest and is written to a purpose: he presents Archimedes as regarding engineering as beneath the dignity of geometry, a philosophical position that suits Plutarch's argument better than it fits the evidence of the treatises.
The burning mirrors said to have set Roman ships alight appear only in much later sources and are not credible.
Death
He was killed at the sack of Syracuse in 212 BCE, against the orders of the Roman commander Marcellus. The versions differ on the details, which is the usual sign of a story elaborated in transmission.
Cicero reports finding and restoring his neglected tomb near Syracuse when serving as quaestor in Sicily, identifying it by the sphere and cylinder carved on it — the result Archimedes had asked to have marked.