Lila means play or sport, and in Hindu theology it names the idea that God's creating and acting are play rather than labour, need or obligation. The reasoning behind it is precise. A being who lacks nothing cannot act in order to obtain anything, so divine activity cannot be explained by motive in the ordinary sense; what remains is activity done for its own sake, which is what play is.
Its classical warrant is Badarayana's Brahma Sutra 2.1.33, which meets the objection that a complete being would have no reason to create by answering that creation is mere play, as is seen in ordinary life. The commentators take the argument up in their own directions. In Advaita Vedanta, lila stands alongside maya as an account of a world that is fully real at its own level while not qualifying or constraining Brahman. In the theistic traditions it becomes warmer and more central: Vaishnava theology, and especially the Gaudiya tradition following Chaitanya, treats Krishna's activities as the paradigm of divine play, the childhood mischief, the stealing of butter, and above all the rasalila with the cowherd women of Vrindavan described in the tenth book of the Bhagavata Purana. Here lila is not merely an explanation of why God acts but an invitation: God plays, and devotion is entry into the play rather than the discharge of an obligation. Drama, dance, poetry and the annual Ramlila performances of northern India enact it directly, and the aesthetic theory of rasa is brought in to describe what the devotee experiences.
Two things about the doctrine deserve notice. It dissolves rather than answers the question of why God created, on the ground that the question presupposes a purpose the tradition denies God needs. And it has been criticised, from within and without, as making the suffering of the world a divine amusement, an objection the traditions answer variously, by distinguishing play from indifference, by insisting that God enters the play and is affected within it, and by reading lila together with karma so that what befalls a person is not arbitrarily assigned. In Shaiva traditions the corresponding image is the dance of Shiva Nataraja, whose five acts include creation, maintenance and dissolution, and which is likewise motion that is not work.