  Bounds and other relations involving variables connected with Carmichael
numbers are reviewed and extended. Families of numbers or individual numbers
attaining or approaching these bounds are given. A new algorithm for finding
three-prime Carmichael numbers is described, with its implementation up to
$10^{24}$. Statistics relevant to the distribution of three-prime Carmichael
numbers are given, with particular reference to the conjecture of Granville and
Pomerance in [A.Granville and C.Pomerance, Two contradictory conjectures
concerning Carmichael numbers, Math. Comp. 71 (2001), 883-908].
