In this talk I will present recent work on multiple timescale dynamical systems displaying complex oscillations with both slow processes and bursts. After reviewing the essential definitions and results about canards in planar and three-dimensional systems, I will focus on two examples of canard-induced spike-adding phenomena in bursting systems. The first one corresponds to square-wave bursting with a slowly-varying input current. Starting from the Hindmarsh-Rose burster, I will explain how the number of spikes per burst can vary through sharp transitions referred to as spike-adding canard explosions, when the applied current I is considered as a bifurcation parameter. Considering a slow evolution for I induces in the resulting four-dimensional model a dynamic passage through the spike-adding regime, which is characterised by ‘Mixed-Mode Bursting Oscillations’ (MMBOs). MMBOs display sub-threshold oscillations, which appear in between the bursts of the original Hindmarsh-Rose model. These small-amplitude oscillations are controlled by so-called folded-node canards. I will give an example of experimental data for which this modelling framework is well suited. The second example will allow me to revisit parabolic bursting, a type of bursting that also requires four state variables, two fast and two slow, from the viewpoint of canards. Focusing on two examples – a conductance-based model called the Plant model and a new polynomial caricature of it –, I will show that the number