Magnetic fields are ubiquitous in the universe and induce fundamental astrophysical processes. It is well known that magnetic fields are important, but their origin and evolution are poorly understood. As a fundamental problem of magnetohydrodynamic (MHD) physics, turbulent dynamo accounts for the growth and maintenance of the cosmic magnetism in most astrophysical settings. However, for decades studies of turbulent dynamo were limited to its linear kinematic regime, which in most cases is of little astrophysical relevance. We have formulated the theory of turbulent dynamo that passes tests of modern turbulent dynamo simulations. It includes the astrophysically significant regimes, such as the dynamo regime at a low ionization fraction and the nonlinear dynamo regime. I will introduce our turbulent dynamo theory and its applications in a wide range of astrophysical conditions to e.g., studying magnetic fields during the first star formation and magnetic field amplification in supernova remnants.