The hazard function is the probability of an event per unit of time for ever smaller time intervals. It has applications to a number of industries including drug development, engineering, finance, insurance and commerce to name a few. We focus on clinical trials, but more specifically, within the therapeutic area of oncology. Here, the hazard function is an important measure that can quantify the changes to the risk of mortality or cancer over time. It is a common and important tool for clinical trial practitioners. In this paper, we develop new non-parametric procedures for testing cumulative hazard functions. From the asymptotic properties of the Kaplan-Meier estimators, we propose procedures that construct test statistics for different tests of hypotheses, including testing if a cumulative hazard function follows a partially known-form hazard, and testing the proportional hazards assumption between two independent samples. Our testing approaches are very flexible since they allow us to choose the testing period and to specify any partially known-form distribution. In addition, the approximate asymptotic distributions of the test statistics are derived under both the null hypothesis and the alternative hypothesis, respectively. Extensive simulation studies show that the proposed procedures enjoy a reasonable Type-I error control and good statistical power under different censoring scenarios. The proposed methodology is further applied to examine the gender-specific mortality hazard rates for young adults with acute myeloid leukemia using the SEER database.