This paper structurally characterizes the complexity of ranking. A set is P-rankable if there is a polynomial time computable function $f$ so that for all $x,f(x)$ computes the number of elements of $A$ that are lexicographically $\le x$, i.e., the rank of $x$ with respect to $A$. We'll say a class $C$ is P-rankable if all sets in $C$ are P-rankable. Our main results show that with the same certainty with which we believe counting to be complex, and thus with at least the certainty with which we believe P $\ne$ NP, we may believe that P has no ranking functions of any type - uniform, strong, weak, or approximate. We show that: P and NP are equally likely to be P-rankable. P is P-rankable if and only if $P\; =\; P^\{\#P\}$. This extends important work of Blum and Sipser [Sip85]. Even weak variations of P-ranking are hard if $P\; \backslash neq\; P^\{\#P\}$. PSPACE is P-rankable if and only if P = PSPACE. If P has small ranking circuits, then it has small ranking circuits of relatively low complexity. If P has small ranking circuits then the power of counting falls into the polynomial hierarchy (i.e., $P^\{\#P\}\; \backslash subseteq\; \backslash (\backslash sum\_\{2\}^\{p\})\backslash \; =\; PH$). P/poly, the class of sets with small circuits is not P-rankable. P/poly has small ranking circuits if and only if $P^\{\#P\}$/poly = $P^\{\#P/poly\}$ = P/poly. If P is rankable, then P/poly has small ranking circuits. This links the ranking complexity of uniform and nonuniform classes. The ranks of some strings in easy sets are of high relative time-bounded Kolmogorov complexity unless P = $P^\{\#P\}$. It follows that even approximate ranking is hard unless P = $P^\{\#P\}$. This partially resolves a question posed by Sipser [Sip85, pp. 447-448].