  Let $\mathfrak a$ denote an ideal of a local ring $(R, \mathfrak m).$ Let $M$
be a finitely generated $R$-module. There is a systematic study of the formal
cohomology modules $\varprojlim \HH^i(M/\mathfrak a^nM), i \in \mathbb Z.$ We
analyze their $R$-module structure, the upper and lower vanishing and
non-vanishing in terms of intrinsic data of $M,$ and its functorial behavior.
These cohomology modules occur in relation to the formal completion of the
punctured spectrum $\Spec R \setminus V(\mathfrak m).$ As a new cohomological
data there is a description on the formal grade $\fgrade(\mathfrak a, M)$
defined as the minimal non-vanishing of the formal cohomology modules. There
are various exact sequences concerning the formal cohomology modules. Among
them a Mayer-Vietoris sequence for two ideals. It applies to new connectedness
results. There are also relations to local cohomological dimensions.
