  We propose a new approach to the fermion sign problem in systems where there
is a coupling $U$ such that when it is infinite the fermions are paired into
bosons and there is no fermion permutation sign to worry about. We argue that
as $U$ becomes finite fermions are liberated but are naturally confined to
regions which we refer to as {\em fermion bags}. The fermion sign problem is
then confined to these bags and may be solved using the determinantal trick. In
the parameter regime where the fermion bags are small and their typical size
does not grow with the system size, construction of Monte Carlo methods that
are far more efficient than conventional algorithms should be possible. In the
region where the fermion bags grow with system size, the fermion bag approach
continues to provide an alternative approach to the problem but may lose its
main advantage in terms of efficiency. The fermion bag approach also provides
new insights and solutions to sign problems. A natural solution to the "silver
blaze problem" also emerges. Using the three dimensional massless lattice
Thirring model as an example we introduce the fermion bag approach and
demonstrate some of these features. We compute the critical exponents at the
quantum phase transition and find $\nu=0.87(2)$ and $\eta=0.62(2)$.
