and
nonabelian homological algebra
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The Leray spectral sequence is the special case of the Grothendieck spectral sequence for the case where the two functors being composed are a push-forward of sheaves of abelian groups along a continuous map $f : X \to Y$ between topological spaces or more generally the direct image of a morphism of sites, followed by the push-forward $Y \to *$ to the point – the global section functor. This yields a spectral sequence that computes the abelian sheaf cohomology on $X$ in terms of the abelian sheaf cohomology on $Y$.
Let $X, Y$ be suitable sites let and $f : X \to Y$ be a morphism of sites. Let $\mathcal{C} = Ch_\bullet(Sh(X,Ab))$ and $\mathcal{D} = Ch_\bullet(Sh(Y,Ab))$ be the model categories of complexes of sheaves of abelian groups. The direct image $f_*$ and global section functor $\Gamma_Y$ compose to $\Gamma_X$:
Then for $A \in Sh(X,Ab)$ a sheaf of abelian groups on $X$ there is a cohomology spectral sequence
that converges as
and hence computes the abelian sheaf cohomology of $X$ with coefficients in $A$ in terms of the cohomology of $Y$ with coefficients in the derived direct image of $A$.
Lecture notes include
Textbook accounts with an eye specifically towards étale cohomology
Günter Tamme, section I 3.7 of Introduction to Étale Cohomology
James Milne, section 11 of Lectures on Étale Cohomology