Zeta-function regularization is arguably the most elegant of the four major regularization methods used for quantum fields in curved spacetime, linked to the heat kernel and spectral theorems in mathematics. The only drawback is that it can only be applied to Riemannian spaces (also called Euclidean spaces), whose metrics have a ++++ signature, where the invariant operator is of the elliptic type, as opposed to the hyperbolic type in pseudo-Riemannian spaces (also called Lorentzian spaces) with a −+++ signature. Besides, the space needs to have sufficiently large symmetry that the spectrum of the invariant operator can be calculated explicitly in analytic form. In the first part we define the zeta function, showing how to calculate it in several representative spacetimes and how the zeta-function regularization scheme works. We relate it to the heat kernel and derive the effective Lagrangian from it via the Schwinger proper time formalism. In the second part we show how to obtain the correlation function of the stress-energy bitensor, also known as the noise kernel, from the second metric variation of the effective action. Noise kernel plays a central role in stochastic gravity as much as the expectation values of stress-energy tensor do for semiclassical gravity.