A sequence of primes $p_1, ..., p_k$ is called a prime chain if $p_j | (p_{j+1}-1)$ for each $j$; e.g. 3, 7, 29, 59.  We will discuss problems about counting prime chains with certain properties, and about the existence of prime chains with various properties.  The Pratt tree for a prime p is the tree with root node p and below p are the Pratt trees of the odd prime factors of $p_1$.Example: 79 ................................../\ ................................3  13 ........................................\ ........................................3 We are concerned with the normal and extremal behavior of the depth of such trees.