The heart of a comet is a cosmic dirty snowball nucleus typically a few kilometres across. We know this because three spacecraft have visited comets in the last twenty years and cameras on board have sent images back to Earth.
The European Space Agency’s Giotto spacecraft flew within 600 km of the nucleus of what's popularly known as "Halley's comet" - officially comet 1P/Halley - in March 1986. The dark nucleus reflected only about four per cent of the light that fell on it. The nucleus was about 15.3 kilometres long and 7 km wide, shaped like a very large smooth potato. Over fifteen years later, in September 2001, the NASA Deep Space 1 craft flew past comet 19P/Borrelly. This nucleus was 8 km long and 3.2 km wide. In January 2004, NASA’s Stardust mission passed the 5.5 x 4.0 x 3.3 km nucleus of 81P/Wild 2. Halley, Borrelly and Wild 2 are reasonably bright comets, but not as bright as the great comet Hale-Bopp in April 1997. This was the brightest comet of the last century, simply because its huge nucleus was 35 km across. More normal comets are smaller than the four we have mentioned so far.
Unlike the planets, on their nearly circular orbits around the Sun, comets have highly elliptical paths. Take Halley. It passes the Sun between the orbits of Mercury and Venus about every 76 years. It was clearly seen in 1986, 1910, 1835, 1759, 1682, 1607, 1531... . At the far end of its orbit Halley is 35 times further away from the Sun than Earth, out in the deep freeze of space beyond Neptune.
Beyond the asteroid belt the cometary nucleus is very cold and inactive. But as the comet approaches the Sun the snow just below the fragile dusty nucleus surface starts to warm up; and when the temperature gets above about 220 Kelvin (Minus 53 Celsius) the ice starts to be converted into gas. The pressure exerted by this escaping gas pushes dust particles away from the nucleus and it becomes surrounded by a large spherical dusty-gassy cloud, known as a coma. These comae can easily be 100,000 km across. Pressure exerted by both sunlight and the solar wind pushes gas and dust away from the coma producing two huge tails. Near the Sun some cometary tails are well over 100,000,000 km long. These can be very startling sights. This activity is extremely useful because it makes the comets easy to see even though their central nuclei are rather small.
The comets that are easy to see are dying. Take Halley again. When it last passed the Earth’s orbit it was losing about 8 x 1029 water molecules every second (that's an 8 with twenty-nine zeros after it). The gas and dust is moving away from the nucleus very quickly, and even at the centre of this active coma the gas density is half a million times lower than the gas at the Earth’s surface. No wonder you can see right through it.
Comet Halley lost a total of 5 x 1011 kg at the last apparition. The Giotto spacecraft showed that only about 10 percent of the nucleus surface was actively emitting gas and dust, and this mass loss is equivalent to that active area retreating 24 m. Active areas are non-permanent. Eventually they ‘turn off’, as a deep dust layer builds up and stops the solar heat getting through to the underlying snows. Activity breaks out in other regions. If this didn’t happen, mass loss would excavate deep pits. This isn't seen to be happening. The surface of Wild 2 goes up and down by a maximum of about 200 m. Maybe this tells us that an active area only lasts for about 10 to 30 apparitions. The migration of active areas stops the nucleus shape becoming too ‘extreme’. It stays between ‘sphere’ and ‘potato’, and rarely moves off in the direction of ‘banana’ or ‘runner bean’. If it did you would expect comets to break-up into more ‘circular’ bits.
The mean radius of Halley’s nucleus is 5300 m. And averaged over the whole surface this comet loses a layer 2.4 m thick every time it passes through the inner solar system. If you divide 5300 by 2.4, you can work out how long Halley has to survive - or, put scientifically: after 5300 / 2.4 apparitions (assuming that the orbit does not change with time) the comet will have completely decayed away. So in 2200 orbits, a mere 170,000 years, Halley’s Comet will have gone.
This high rate of mass loss means that the nucleus surface is only a few thousand years old. Considering the state of flux of asteroids it is highly unlikely that any cometary surface has been cratered by asteroidal impact.