  An isoparametric family in the unit sphere consists of parallel isoparametric
hypersurfaces and their two focal submanifolds. The present paper has two
parts. The first part investigates topology of the isoparametric families,
namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability,
as well as the Lusternik-Schnirelmann category. This part extends substantially
the results of Q.M.Wang in \cite{Wa88}. The second part is concerned with their
curvatures, more precisely, we determine when they have non-negative sectional
curvatures or positive Ricci curvatures with the induced metric.
