The tremendous progress of cosmological observations in the last few years is showing us with greater and greater confidence that in our past the universe underwent a phase of cosmic acceleration, called inflation. After reviewing what inflation is and what its main predictions are, I will show that in some cases the inflationary phase can be never-ending and become eternal. This fact could be related to another very important cosmological observation: the current acceleration of the universe. This represents one of the greatest mysteries in our present understanding of nature. I will show how this connection is possible, presenting at the same time other alternative explanations.