A bifurcation is a qualitative (topological) change of the phase portrait under variation of system parameters. For example, the coalescence and disappearance of equilibria or fixed points, or a change of their stability character are bifurcations. In this chapter, we study local bifurcationsBifurcationlocal, i.e. those which happen in small neighbourhoods of equilibria or fixed points. We focus on dynamical systems depending on one parameter. Local bifurcations happen at the (exceptional) parameter values for which the steady state under consideration lacks hyperbolicity. Our strategy will be to study in this chapter the simplest local bifurcations in generic one-parameter systems in state spaces of the lowest possible dimension. The dimension is determined by the number of critical eigenvaluesEigenvaluecritical at the bifurcating equilibrium (fixed point), i.e. those eigenvalues which make the point nonhyperbolic. The loose meaning of “generic” is “as a rule”, while the precise meaning depends on the context. The general idea is that “generic” excludes extra degeneracies that are exceptional in the situation that we consider. In Chap.  6 , we will show how to apply the obtained results in general n-dimensional situations.