Cancer patients often experience anemia due to the disease itself, chemotherapy, or both. Quality of life is also affected, due in part to the fatigue associated with anemia. Previous studies with epoetin alfa have suggested that achieving a higher hemoglobin level may improve quality of life and help patients live longer. This is a randomized, double-blind, placebo-controlled, multicenter study to assess the effect of treatment with epoetin alfa in reducing or preventing anemia and transfusions in patients receiving non-platinum chemotherapy for non-myeloid cancers. The study also aims to determine whether changes in erythropoietin and hemoglobin levels after 2 weeks, serum ferritin (iron) levels after 2 weeks and changes in hemoglobin and developing red blood cells after either 2 or 4 weeks predict responsiveness to epoetin alfa. There will be 2 treatment groups: one group will receive subcutaneous epoetin alfa injections 3 times per week (starting at 150 units per kilogram, adjusted if needed to a maximum of 300 units per kilogram) and the other group will receive an equal volume of matching subcutaneous placebo. Treatment is to continue for 12 to 24 weeks (3 to 6 chemotherapy cycles), plus 4 weeks post-chemotherapy. Patients in the study will be observed a year after the study ends for survival data. The primary measure of effectiveness will be determined by the number of patients who are transfused, depending on primary tumor type (solid or blood) and level of hemoglobin (above or below 10.5 grams per deciliter). Additional measures of effectiveness include the change in hemoglobin, hematocrit, and developing red blood cells from the start of the study until study completion, and the change in quality of life. Safety evaluations (incidence and severity of adverse events, laboratory tests, vital signs and physical examinations) will be performed throughout the study. The hypothesis of the study is that epoetin alfa will be superior to placebo in reducing the need for transfusions and in improving anemia and quality of life. 150 units per kilogram of epoetin alfa (or placebo) 3 times weekly subcutaneously for 4 weeks, then continuing this dose or 300 units per kilogram, depending on developing red blood cell counts and/or hemoglobin. Study duration is 12 to 24 weeks (3 to 6 cycles) plus 4 weeks post-chemotherapy.