We study topological phases in orthorhombic perovskite iridium (Ir) oxide superlattices grown along the crystallographic axis. Bilayer Ir oxide superlattices display topological magnetic insulators exhibiting quantized anomalous Hall effects due to strong spin-orbit coupling of Ir orbitals and electronic correlation effects. We also find a valley Hall insulator with counterpropagating edge currents from two different valleys and a topological crystalline insulator with edge states protected by the crystal lattice symmetry based on stacking of two layers. In a single-layer superlattice, a topological insulator can be realized, when a strain field is applied to break the symmetry of a glide plane protecting the Dirac points. It turns into a topological magnetic insulator in the presence of magnetic ordering and/or in-plane magnetic field. We discuss essential ingredients for these topological phases and experimental signatures to test our theoretical proposals.