  Voevodsky's derived category of motives is the main arena today for the study
of algebraic cycles and motivic cohomology. In this paper we study whether the
inclusions of three important subcategories of motives have a left or right
adjoint. These adjoint functors are useful constructions when they exist,
describing the best approximation to an arbitrary motive by a motive in a given
subcategory. We find a fairly complete picture: some adjoint functors exist,
including a few which were previously unexplored, while others do not exist
because of the failure of finite generation for Chow groups in various
situations. For some base fields, we determine exactly which adjoint functors
exist.
