We present results on charmonia correlators and spectral functions at finite temperature from lattice QCD. We use anisotropic lattice and Fermilab action at several lattice spacings. The use of anisotropic action allows us to have many data points in the time direction, up to $N_t=44$ above deconfinement transition. We confirm earlier findings that 1S state survive in the plasma up to temperatures $1.5T_c$, while the 1P states are dissolved at temperatures close to $T_c$. The charmonia correlators in the vector channel are studied further in detail both in lattice simulations and analytically. Using effective Langevin theory for charm quarks at $p, \omega << T$ we identify the very low energy contribution to the vector spectral functions. It is shown how the low energy part of the spectral function depends on the heavy quark diffusion coefficient $D$ and to what extent lattice calculation of the vector correlator can constrain the value of the heavy quark diffusion coefficient.