The width of a Resolution proof is defined to be the maximal number of
literals in any clause of the proof. In this paper we relate proof width
to proof length (=size), in both general Resolution, and its tree-like
variant. The following consequences of these relations reveal width as a
crucial ``resource'' of Resolution proofs.

In one direction, the relations allow us to give simple, unified proofs
for almost all known exponential lower bounds on size of resolution
proofs, as well as several interesting new ones. They all follow from
width lower bounds, and we show how these follow from natural expansion
property of clauses of the input tautology.

In the other direction, the width-size relations naturally suggest a
simple dynamic programming procedure for automated theorem proving - one
which simply searches for small width proofs. This relation guarantees
that the running time (and thus the size of the produced proof) is at most
quasi-polynomial in the smallest tree-like proof. The new algorithm is
never much worse than any of the recursive automated provers (such as DLL)
used in practice. In contrast, we present a family of tautologies on which
it is exponentially faster.