We study the cyclicity of limit periodic sets that occur in families of vector fields of slow–fast type. The limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point. At this turning point a stability change takes place: on one side of the turning point the dynamics point strongly towards the curve of singularities; on the other side the dynamics point away from the curve of singularities. The presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point, i.e. orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve. All existing results deal with a non-zero slow movement, permitting a good estimate of the cyclicity by considering the slow-divergence integral along the curve of singularities. In this thesis we study what happens when the slow dynamics exhibit singularities. In particular, our study includes the cyclicity of the slow–fast two-saddle cycle, formed by a regular saddle connection (the fast part) and a part of the curve of singularities (the slow part). We see that the relevant information is no longer merely contained in the slow-divergence integral. This thesis concerns the study of the cyclicity of limit periodic sets in a quite general normal; MARGIN: 0cm 0cm 0pt; mso-layout-grid-align: none" slow–fast vector fields on a 2-manifold M . We are interested in families of vector fields X ? (possibly depending on other parameters as well) where the unperturbed ‘fast’ vector field X 0 has a curve of singular points , called a critical curve. We call a point p on normally attracting (respectively, normally repelling) when DX 0( p ) has a strictly negative (respectively, strictly positive) eigenvalue corresponding to an eigendirection not tangent to . When has both normally attracting points and normally repelling points, X 0 may have orbits connecting two such points. Let F be such a fast orbit so that the ? -limit and ? -limit lie on . We study the limit periodic set L F formed by F and the part of going from the ? -limit of F to the ? -limit of F . Of course the non-zero eigenvalue of DX 0( p ) must bifurcate along this part of ; assume that this happens at a unique point p * called a turning point. (One also says normal hyperbolicity of X 0 is lost in p * .) In such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits; such orbits are called canard cycles.