If computational complexity is the study of what makes certain computational

problems inherently difficult to solve, an important contribution of

descriptive complexity in this regard is the separation it provides between

the specification of a decision problem and the structure against which this

specification is checked. The formalisation of these two aspects leads to

tools for studying them as sources of complexity, and on the one hand leads to

results in the characterisation of complexity classes and on the other elates

to parameterized complexity. In these notes accompanying the invited talk,

some definitions and results are presented leading to recent work on the

characterisation of polynomial time and on the parameterized complexity of

first-order logic on restricted graph classes.