We introduce notions of compactness and weak compactness for multilinear maps
from a product of normed spaces to a normed space, and prove some general
results about these notions. We then consider linear maps $T:A\rightarrow B$
between Banach algebras that are ``close to multiplicative'' in the following
senses: the failure of multiplicativity, defined by $S_T(a,b)=T(a)T(b)-T(ab)$
$(a,b\in A)$, is compact [respectively weakly compact]. We call such maps
cf-homomorphisms [respectively wcf-homomorphisms]. We also introduce a number of
other, related definitions. We state and prove some general theorems about these
maps when they are bounded, showing that they form categories and are closed
under inversion of mappings and we give a variety of examples. We then turn our
attention to commutative $C^*$-algebras and show that the behaviour of the
various types of ``close-to-multiplicative'' maps depends on the existence of
isolated points in the maximal ideal space. Finally, we look at the splitting of
Banach extensions when considered in the category of Banach algebras with
bounded cf-homomorphisms [respectively wcf-homomorphisms] as the arrows. This
relates to the (weak) compactness of 2-cocycles in the Hochschild-Kamowitz
cohomology complex. We prove ``compact'' analogues of a number of established
results in the Hochschild-Kamowitz cohomology theory.