In 1986, Puri and Ralescue introduced a new concept of fuzzy random variable and its expectation in order to treat nonstatistical imprecise data by the statistical method. The purpose of this paper lies in further development of the theory of fuzzy random variables by introducing the conditional expectations and martingales of fuzzy random variables. To do so, we define the addition, multiplication by real-valued measurable function and another one operation and investigate their properties in terms of the expectation. We introduce the conditional expectation of the fuzzy random variables and investigate its properties under the above operations. Furthermore, we show that the properties which hold for ordinary conditional expectations are true for the conditional expectations of fuzzy random variables with some modifications. Fuzzy martingales are also introduced and their convergence theorems for closed martingales are established for both increasing parameter and decreasing parameter cases. We show that the uniformly integrable martingale of fuzzy random variables is represented by a closed martingale.