Motivated by the recent wide applications of non-convex smooth games, we provide a unified approach to "local optimal" points in such games, which includes local Nash equilibria, local minimax points (Jin et al. 2019) and the more general local robust points. To understand these definitions further, we study their corresponding first- and second-order necessary and sufficient conditions and find that they all satisfy stationarity. This motivates us to analyze the local stability of several popular gradient algorithms near corresponding local solutions. Our results indicate the necessity of new algorithms and analysis. As a concrete example, we give the exact existence conditions of local (global) minimax points and local robust points for quadratic games, and demonstrate their many special properties.