Perforated graphene and other perforated two-dimensional materials can be used in hemodialysis membranes and blood filtration membranes for selective removal of components from blood in vivo and ex vivo. The membranes are useful in hemodialysis and hemofiltration techniques to provide improved patient care. Hemodialysis systems can include a hemodialysis membrane formed from perforated graphene or another perforated two-dimensional material disposed upon a porous support structure. Hemofiltration systems can include one or more and preferably two or more blood filtration membrane formed from perforated graphene or another perforated two-dimensional material disposed upon a porous support structure. Methods for performing hemodialysis can involve exposing blood from a patient to a hemodialysis membrane formed from a perforated two-dimensional material. Ex vivo dialysis techniques can be performed similarly. Methods for filtration of blood can involve passing blood through one or more filter membranes or through a plurality of sequential filter membranes.