  We show that $J-$ stability is open and dense in natural families of
meromorphic maps of one complex variable with a finite number of singular
values, and even more generally, to finite type maps. This extends the results
of Ma\~{n}\'e-Sad-Sullivan for rational maps of the Riemann sphere and those of
Eremenko and Lyubich for entire maps of finite type of the complex plane, and
essentially closes the problem of density of structural stability for
holomorphic dynamical systems in one complex variable with finitely many
singular values. This result is obtained as a consequence of a detailed study
of a new type of bifurcation that arises with the presence of both poles and
essential singularities (namely periodic orbits exiting the domain of
definition of the map along a parameter curve), and in particular its relation
with the bifurcations in the dynamics of singular values. The presence of these
new bifurcation parameters require essentially different methods to those used
in previous work for rational or entire maps.
