Semiclassical gravity (SCG) theory provides a self-consistent description of spacetime dynamics and quantum matter fields through the expectation values of their quantum stress energy tensors. When their fluctuations become large, as could happen in the early universe or late stages of black hole collapse, SCGmay fail to provide an accurate description.The proper framework to address issues involving the dynamics of fluctuations in the matter field and in the metric tensor consistently is stochastic gravity . The fluctuations in the matter field is contained in the noise kernel which is the expectation value of the stress energy bi-tensor acting as a source for the Einstein-Langevin equation, from which one can obtain the induced metric fluctuations in the background spacetime. Stochastic gravity program is currently investigated for metric fluctuations and backreaction in evaporating black holes and cosmological structure formation. We describe recent results in the calculations of the noise kernels of conformally-invariant scalar fields in spacetimes conformal to an ultrastatic spacetime (under quasilocal approximation), in static de Sitter and all conformally-flat spacetimes (exact expressions) as well as for massive quantum scalar field with general coupling in Euclidean R^N and AdS^N spaces via the generalized zeta-function method.