The circumstellar habitable zone (CHZ) is usually described as a sphere surrounding a star (which translates into an annulus in the orbital plane) capable of having planets with surface water. The CHZ is necessary for the development and maintenance of complex life, but it is not sufficient. Here we address the idea that ionizing radiation also has a major determining effect for planets that are either favorable to or inhospitable to complex life as we know it. We discuss high-energy radiation and particles from the bedrock of the Earth, as well as the Sun, and cosmic rays from both Galactic and extra-galactic sources. There are two general kinds of ionizing radiation: 1) high-energy electromagnetic waves like gamma rays, X-rays, and UV radiation, and 2) particle radiation, including radioactivity, from the Earth's crust and cosmic rays. Cosmic rays include high-energy particles from the Sun, and particles from high-energy sources in the Galaxy, like novae, kilonovae, supernovae, microquasars, and gamma-ray bursts. Ultra-high energy cosmic rays originate from extra-galactic sources, like accreting black holes with jets. We discuss the various threats involving ionizing radiation and their effects on complex life. Essentially all galaxies at least as massive as the Milky Way have central supermassive black holes which may often be quiescent, but could occasionally be quite active. We focus here especially on supernovae and gamma-ray bursts, as well as the Galactic cosmic rays associated with these explosions as these are the most prevalent threats constraining complex life in the Milky Way today. We discuss the layers of biosphere protection against cosmic rays including the planet's atmosphere, the ozone layer, the planetary magnetosphere, the stellar wind and its boundary layer outside (known as the astrosphere and astropause), the Galactic magnetic field and the Galactic magnetic wind with its boundary layer (which we call the Galactosphere and Galactopause). In addition, the effects of ionizing radiation on life on Earth's surface and in shallow water are discussed. We summarize the evidence for nearby supernovae in Earth's past and the current mass extinction rates from stellar explosions. Finally, the speculation on an astrophysical cause of the homochirality of organic matter is addressed.