There's a whole slew of methods for dealing with various forms of dependence.
Time series and spatial methods in part deal with issues that relate to the way observations that are "close together" (in time, or space) may tend to be more strongly dependent than more distant observations.
Methods like principal components, factor analysis and in a much more general sense, copulas, may be used to describe or model a variety of kinds of multivariate dependence between variates.
Random effects/mixed effects models may be understood as a way to model some forms of dependence that could be seen as due to shared dependence on components of variation that are not explicit explanatory variates (a simple example would be the dependence due to shared class-memberships that shift the mean, for example when all the measurements in one class are on the same individual, who has individual variation that makes all their measurements tend to be more alike than they are like measurements of other individuals; another example would be student marks that tend to more alike with each other, intra-classroom than with other-student inter-classroom marks because of the effect of a shared teacher)
I've barely scratched the surface here, so see this as a list of a few examples of modelling forms of dependence than the whole gamut.