In this talk, I will discuss Deligne-Hitchin-Simpson's twistor space related to the moduli space of λ-connections on a smooth projective curve with regular or irregular singularities of fixed (generalized) local exponents. All known examples of Painlevé equations can be obtained by isomonodromic deformations of linear connections with singularities. Therefore the phase spaces of Painlevé equations are given by families of moduli spaces of connections with singularities of fixed generalized local exponents. (I will briefly review about the classical Painlevé equations and the Riemann-Hilbert correspondences related to them.) By using the explicit description of moduli spaces of λ-connections, we can explain how these Painlevé equations are degenerating to the Hitchin's integrable systems in the twistor spaces. As an example, we will give an explicit calculation of the degeneration of Painlevé VI equations to Hitchin system.