Many natural computational problems have input consisting of two or mo
re parts, one of which may be considered a parameter. For example, the
re are many problems for which the input consists of a graph and a pos
itive integer. A number of results are presented concerning parameteri
zed problems that can be solved (uniformly with respect to the paramet
er) in complexity classes below P, given a single word of advice for e
ach parameter value. Different ways in which the word of advice can be
employed are considered, and it is shown that the class FPT of tracta
ble parameterized problems (the parameterized analog of P) has interes
ting and natural internal structure.