A closed 1-form, viewed locally, is a smooth function up to an additive constant. The global structure of a closed 1-form depends on its de Rham cohomology class. In 1981, S. P. Novikov started studying closed 1-forms with Morse type zeros; he constructed a generalization of the Morse theory giving lower bounds on the number of zeros of Morse closed 1-forms lying in a prescribed cohomology class. In the talk I will discuss a more recent theory which studies zeros of closed 1-forms without assuming that they are Morse type. This theory relates some new homotopy invariants of manifolds to interesting dynamical properties of flows. In particular, it allows to predict (using homotopical information) existence of homoclinic cycles in dynamical systems. I will also discuss the notion of a Lyapunov 1-form generalizing the classical notion of a Lyapunov function.