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	Abstract
1	Overview
2	From Semiclassical to Stochastic Gravity
	2.1	The importance of quantum fluctuations
3	The Einstein-Langevin Equation: Axiomatic Approach
	3.1	Semiclassical gravity
	3.2	Stochastic gravity
4	The Einstein-Langevin Equation: Functional Approach
	4.1	Influence action for semiclassical gravity
	4.2	Influence action for stochastic gravity
	4.3	Explicit form of the Einstein-Langevin equation
5	Noise Kernel and Point-Separation
	5.1	Point separation
	5.2	Stress-energy bi-tensor operator and noise kernel
6	Metric Fluctuations in Minkowski Spacetime
	6.1	Perturbations around Minkowski spacetime
	6.2	The kernels in the Minkowski background
	6.3	The Einstein-Langevin equation
	6.4	Correlation functions for gravitational perturbations
	6.5	Discussion
7	Structure Formation
	7.1	The model
	7.2	The Einstein-Langevin equation for scalar metric perturbations
	7.3	Correlation functions for scalar metric perturbations
	7.4	Discussion
8	Black Hole Backreaction
	8.1	The model
	8.2	CTP effective action for the black hole
	8.3	Near flat case
	8.4	Near horizon case
	8.5	The Einstein-Langevin equation
	8.6	Discussions
9	Concluding Remarks
10	Acknowledgements
	References
	Footnotes

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