1.609344, exactly
The international mile is exactly 1,609.344 metres — a consequence of the mile being 5,280 feet and the foot being exactly 0.3048 m. So the conversion is exact, and the familiar 1.6 shortcut is a rounding that costs about 0.6%.
The Fibonacci trick
Consecutive Fibonacci numbers approximate the ratio well, because the golden ratio (1.618) is close to 1.609. So 5 km is about 3 miles, 8 km about 5 miles, 13 km about 8 miles and 21 km about 13 miles — each within about half a percent. It is a genuinely useful piece of mental arithmetic for runners, and it is why a half marathon at 21.1 km lands on 13.1 miles.
Pace is the inverse conversion
Distance and pace convert in opposite directions, which is where mistakes creep in. A longer distance in miles means a bigger number of kilometres, but a pace per mile is a bigger number of minutes than the same pace per kilometre — because you are covering more ground per unit. Running 5:00/km is running 8:03/mile. Multiply pace per km by 1.609344 to get pace per mile.
Nautical miles are a different unit again
A nautical mile is 1,852 metres exactly, originally one minute of latitude along a meridian. It is about 15% longer than a statute mile, which is why aviation and marine speeds in knots look slower than the equivalent mph. The page shows nautical miles alongside so the three do not get confused.