As noted in the answer by Tom Nathe, the galactic core is too tightly packed with stars to have conditions that would be conducive to life. The proximity of stars to each other means that the orbits any planets around these stars would be disturbed by gravitational interactions with other stars, and thus would likely not have stable orbits.
Because the core of the galaxy is not usually thought to be a good place for habitable worlds, some astrobiologists identify a “galactic habitable zone” (GHZ), which is an habitable zone for the entire galaxy, analogous to the habitable zone around a given star, which latter is then referred to as a “circumstellar habitable zone” (CHZ). Any planetary system outside the GHZ would be considered an unlikely place to find life.
Globular clusters—small mini-galaxies of thousands to millions of stars that orbit larger galaxies—are similarly closely packed with stars and so pose problems for planetary habitability. A recent paper examined this: “Habitability in the Omega Centauri Cluster” by Stephen R. Kane and Sarah J. Deveny.
Another problem with globular clusters—a problem from the perspective of being clement to life—is that the processes of galactic ecology that work in the main body of the galaxy do not work in globular clusters. Galactic ecology is what we call the recycling of material from stars that explode in a supernova, scattering their remnants, which are then later incorporated into later generations of stars, which as a consequence have a higher level of heavier elements (both due to nucleosynthesis while the former star was fusing elements in its core, as well as new elements created by the supernova event itself). Planetary systems that incorporate more heavy elements (i.e., are higher in metallicity) are likely to be more minerologically and hence more geologically complex, and it is likely that geological complexity plays a role in the emergence of life.
An important caveat to the above: it should be observed that conventional conceptions of habitability have been questioned recently as we have learned that moons in our solar system (and probably also planets elsewhere in our galaxy) have large subsurface oceans under kilometers of ice exposed to the cold and vacuum of space. It is possible that life could arise in subsurface oceans, in which case the requirement of a planet being in a CHZ where liquid water could be found on its surface may be too narrow a criterion for searching for life and for the definition of a habitable zone.
If we reconfigure the idea of a habitable zone to allow for life in subsurface oceans, and perhaps also to allow for kinds of life radically different from life on Earth, the GHZ may be much larger than in the illustration above, and it may, in fact, include all regions of our galaxy.