Patients with cancers of the blood often develop low blood cell counts either as a consequence of the disease or the treatment by chemotherapy or stem cell transplantation. Platelet transfusions are commonly given to raise any low platelet count and reduce the risk of clinical bleeding (prophylaxis) or stop active bleeding (therapy). But recent research has indicated that many patients continue to experience bleeding, despite the use of platelet transfusions. Tranexamic acid is a type of drug that is called an antifibrinolytic. These drugs act to reduce the breakdown of clots formed in response to bleeding. These drugs have been used widely in both elective and emergency surgery and have been shown to decrease blood loss and the use of red cell transfusions. The purpose of this study is to test whether giving tranexamic acid to patients receiving treatment for blood cancers reduces the risk of bleeding or death, and the need for platelet transfusions. Patients will be randomised to receive tranexamic acid (given intravenously through a drip, or orally) or a placebo. We will measure the rates of bleeding daily using a short structured assessment of bleeding, and we will record the number of transfusions given to patients.