Additional Study Information: The immune system can be useful in attacking cancer cells, but there are ways that the tumor can reduce or prevent the activity of the immune system. This is called immune suppression. Blocking immune suppression can allow the immune cells to attack again. There are several types of immune suppression. One example is adenosine that is made within the tumor and binds to a receptor called A2AR. Laboratory tests have shown that blocking A2AR can improve the function of immune cells, which can lead to anti-tumor responses. The study drug, AZD4635, is an oral medicine that blocks adenosine signals through A2AR. You are being asked to participate in this research study because you have a solid tumor that does not respond to any known treatment. The purpose of the study is to investigate the safety and tolerability of AZD4635 in patients with cancer. Some subjects may have previously received immune checkpoint inhibitor therapy (for example, nivolumab, pembrolizumab, atezolizumab, or avelumab) and either progressed or responded and then stopped responding.We are testing what dose and schedule of AZD4635 can be given and whether it can prevent tumors from occurring or growing bigger. In Part 1a of this study, AZD4635 was first tested alone and then in combination with durvalumab. While this is the first study using AZD4635, durvalumab has already been given to subjects in many clinical studies. Levels of these drugs in your blood will also be measured, to see how the body accepts and gets rid of them. Samples of your blood and tumor tissue will also be taken to examine for possible cancer markers.