  Incomplete cooperative games generalise the classical model of cooperative
games by omitting the values of some of the coalitions. This allows to
incorporate uncertainty into the model and study the underlying games as well
as possible payoff distribution based only on the partial information. In this
paper we perform a systematic study of incomplete games, focusing on two
important classes of cooperative games: positive and convex games.
  Regarding positivity, we generalise previous results for a special class of
minimal incomplete games to general setting. We characterise non-extendability
to a positive game by the existence of a certificate and provide a description
of the set of positive extensions using its extreme games. The results are then
used to obtain explicit formulas for several classes of incomplete games with
special structures.
  The second part deals with convexity. We begin with considering the case of
non-negative minimal incomplete games. Then we survey existing results in the
related theory of set functions, namely providing context to the problem of
completing partial functions. We provide a characterisation of extendability
and a full description of the set of symmetric convex extensions. The set
serves as an approximation of the set of convex extensions.
  Finally, we outline an entirely new perspective on a connection between
incomplete cooperative games and cooperative interval games.
