The lecture notes of my ESSLLI'01 course are now available in a revised version at http://www.cwi.nl/~kurz/cml-esslli01.html The main aim of this course has been to sketch some current approaches to modal logics for coalgebras (Chapter 4) and to explain the duality of modal and equational logic (Chapter 5). To make the course as self-contained as possible, I have also included brief introductions to applications of coalgebras in computer science, to some categorical constructions on coalgebras, and to modal logic. Alexander Kurz