Tsunami hazard assessments have frequently been conducted by means of numerical models using deterministic inputs, such as the earthquake fault parameters and bathymetry. Some fault parameters of the earthquake, however, can only be estimated probabilistically for a future event. The bathymetry, on the other hand, is surveyed with an imperfect resolution and accuracy, which can lead to errors in the tsunami modeling results. Hence, tsunami assessments before an event are unavoidably uncertain. This thesis aims to describe how the uncertainties of some earthquake fault parameters and bathymetry impact the tsunami assessment uncertainties. The earthquake uncertainties are related with the assumed aleatory nature of the slip distribution and the rupture location, while the bathymetry uncertainties are associated with the lack of data in unsurveyed areas. These uncertain inputs and the corresponding tsunami response are modeled as random elements by adopting a stochastic approach. The uncertain earthquakes of this thesis consider a slip distribution modeled as an homogeneous random field and a rupture location modeled as a random vector. The bathymetry, on the other hand, is modeled as a non-homogeneous Gaussian random field, which is conditional to surveyed data. The generation of samples of the earthquake rupture location is straightforward. Conversely, the generation of samples of the slip distribution and bathymetry are rather complex. By means of a Karhunen-Loeve expansion and a translation model we propose a consistent method for the generation of samples of these uncertain inputs. Unlike other approaches, the Karhunen Loeve expansion generates consistent and accurate samples of non-rectangular random fields. The uncertainties of tsunami hazard assessments are then quantified by means of a Stochastic Reduced Order Model (SROM), which is more accurate than the classic Monte Carlo simulation for a same number of samples. The uncertainty quantification methods developed in this thesis are presented with two illustration cases. In one illustration case we study MW 8.0 earthquakes within a seismogenic region in North Chile. First, we demonstrate that our proposed method generates consistent earthquake and bathymetry samples. Second, we demonstrate that estimates of tsunami assessment uncertainties obtained with SROM are more accurate than estimates obtained with classic Monte Carlo simulations. From sensitivity analyses and comparison with records of the 2014 earthquake tsunami, we also conclude that the probability properties of the analyzed uncertain inputs and other aspects of the tsunami assessment can be relevant sources of uncertainty. In the second illustration case we perform a probabilistic tsunami hazard assessment (PTHA) which assesses earthquakes generated in the Manila Subduction Zone and tsunami responses in Hong Kong, China, and Kao Hsiung, Taiwan. First we demonstrate that our proposed methods can be combined with PTHA to account for the earthquake fault parameters and bathymetry uncertainties. Second, we demonstrate that the earthquakes recurrence model and the tsunami propagation model adopted in the PTHA constitute additional sources of uncertainty, which can be as relevant as the uncertainties of the earthquake fault parameters and bathymetry.