  The notion of almost Gorenstein local ring introduced by V. Barucci and R.
Fr\"oberg for one-dimensional Noetherian local rings which are analytically
unramified has been generalized by S. Goto, N. Matsuoka and T. T. Phuong to
one-dimensional Cohen-Macaulay local rings, possessing canonical ideals. The
present purpose is to propose a higher-dimensional notion and develop the basic
theory. The graded version is also posed and explored.
