In this paper we develop evolutionary strategies for numerical continuation which we apply to scalar and multi-objective optimization problems. To be more precise, we will propose two different methods-an embedding algorithm and a multi-objectivization approach-which are designed to follow an implicitly defined curve where the aim can be to detect the endpoint of the curve (e.g., a root finding problem) or to approximate the entire curve (e.g., the Pareto set of a multi-objective optimization problem). We demonstrate that the novel approaches are very robust in finding the set of interest (point or curve) on several examples.