Today we focus on W-hardness, i.e. we focus on problems that are probably not fixed-parameter tractable (similar to P vs NP). This includes a short introduction to the W-hierarchy.
Since some notions of approximability imply the fixed-parameter tractability of a parameterized version of the problem, W[1]-hardness proofs are an important tool (for example in proving that a problem is unlikely to admit an efficient PTAS).