The existence of Nash equilibrium and sufficient conditions of Levitin-Polyak well-posedness for population games (LP well-posedness, for short) are investigated. Firstly, by taking advantage of an auxiliary optimization problem and Fan-KKM lemma respectively, existence results of Nash equilibrium for population games are obtained. Secondly, LP well-posedness for population games is introduced, and the metric characterization of LP well-posedness in the behavior of approximate solution set for population games is discussed. Finally, the sufficient conditions of LP well-posedness for population games are established. The existence and LP well posedness of Nash equilibrium for group game problems are established under the conditions of upper 0-level closure and quasi convexity. The existence of Nash equilibrium for group game problems is established under weaker conditions. LP well-posedness of Nash equilibrium for group game problems is proposed and sufficient conditions for it is established.