a b s t r a c tLet R be a commutative local noetherian ring, and let L and L ′ be R-modules. We investigate the properties of the functors Tor R i (L, −) and Ext i R (L, −). For instance, we show the following: (a) if L and L ′ are artinian, then Tor R i (L, L ′ ) is artinian, and Ext i R (L, L ′ ) is noetherian over the completion R; (b) if L is artinian and L ′ is Matlis reflexive, then Ext i R (L, L ′ ), Ext i R (L ′ , L), and Tor R i (L, L ′ ) are Matlis reflexive.Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.