  In this paper, we give a systematic construction of ten isomonodromic
families of connections of rank two on $\mathbb{P}^1$ inducing Painlev\'e
equations. The classification of ten families is given by considering the
Riemann-Hilbert morphism from a moduli space of connections with certain type
of regular and irregular singularities to a corresponding catetorical moduli
space of analytic data (i.e., ordinary monodromy, Stokes matrices and links),
which is called the monodromy space. Explicit equations of the monodromy spaces
for ten families are calculated. Moreover, we obtain natural explicit families
of connections for these ten cases and calculate isomonodromic equations which
give Painlev\'e equations of all types.
