We introduce and study clones of partial cofunctions on sets of arbitrary cardinality. In the process, we outline a general Galois theory similar to Pol-Inv, show some elementary results about the essential arity of clones of partial cofunctions, take a closer look at partial idempotent cofunctions, characterize all minimal clones of partial cofunctions, and show that the join of all minimal clones is the full clone (provided that the Axiom of Choice is assumed).