The period (roughly) 1990-today is characterized by a big watershed and branching of cosmology into multiple and hitherto unexpected directions. On one side, the generic chaotic/eternal inflation has provided physical grounds for rather wild speculative ideas about the multiverse: the possibly infinite set of cosmological domains (‘universes’). In order to determine how observed features of our universe are (im)probable in the multiverse context requires application of anthropic reasoning which is still controversial in many circles. On the other side, we encounter applications of other speculative physical theories, like the string/M-theory to cosmology, resulting in unusual hypotheses like those of the pre-Big Bang cosmologies. In this period we have also witnessed the birth of physical eschatology as the true ‘cosmology of the future’. This chapter will attempt a survey of these and related developments, with necessary qualifications which accompany any ongoing, evolving research activity.