In this tutorial paper we survey some of the existing techniques for modelling, analysis and synthesis of asynchronous control circuits. All these methods are based on the use of Petri nets as a tool for describing the behaviour of such circuits. The descriptive power of Petri nets allows them to model a wide range of asynchronous circuit components, whether they are built in the two-phase (micropipeline) or in the four-phase (logic gate based) design styles. We present three different approaches to verification of net-based, and show their relative strengths and weaknesses. We advocate their complementary application for different classes of Petri nets and the properties verified. Two major synthesis approaches are demonstrated using the example of a modulo- N Up/Down counter. The first one is a combination of Petri net level decompositions and syntax-directed translation of nets into circuits. The second one is based on logic synthesis from signal transition graph specifications.