paramaterized complexity looks at the time requirements for problems as a function of both instance size and other measures of the difficulty of an instance. for example, while finding an independent set of size $k$ in a graph is an $np$-complete problem, for each fixed $k$, the problem is clearly solvable in polynomial time through exhaustive search. parameterized complexity answers such questions as: can this problem can be solved in polynomial-time for $k$ a growing function $k(n)$; and, to what extent can the naive $n^{o(k)}$ algorithm be improved? there have been a number of very interesting connections made recently between parameterized complexity, the exponential time hypothesis (that 3sat requires time $2^\omega(n)$) and several other classical notions. i will discuss some of these, giving the necessary background material. if time permits i will also talk about applications to online models.