The covariance matrix of the matter and halo power spectrum and bispectrum are studied. Using a large suite of simulations, we find that the non-Gaussianity in the covariance is significant already at mildly nonlinear scales. We compute the leading disconnected non-Gaussian correction to the matter bispectrum covariance using perturbation theory, and find that the corrections result in good agreement in the mildly nonlinear regime. The shot noise contribution to the halo power spectrum and bispectrum covariance is computed using the Poisson model, and the model yields decent agreement with simulation results. However, when the shot noise is estimated from the individual realization, which is usually done in reality, we find that the halo covariance is substantially reduced and gets close to the Gaussian covariance. This is because most of the non-Gaussianity in the covariance arises from the fluctuations in the Poisson shot noise. We use the measured non-Gaussian covariance to access the information content of the power spectrum and bispectrum. The signal-to-noise ratio, S/N, of the matter and halo power spectrum levels off in the mildly nonlinear regime, k ∼ 0.1 − 0.2 M p c h − 1 k ∼0.1 - 0.2 ,mathrmMpc h^-1 . In the nonlinear regime the S/N of the matter and halo bispectrum increases but much slower than the Gaussian results suggest. We find that both the S/N for power spectrum and bispectrum are overestimated by the Gaussian covariances, but the problem being much more serious for the bispectrum. Because the bispectrum is affected strongly by nonlinearity and shot noise, inclusion of the bispectrum only adds modest amount of S/N compared to that of the power spectrum.