Risk assessment for natural hazards and financial extreme events requires the statistical analysis of extreme events, often beyond observed levels. The characterization and extrapolation of the probability of rare events rely on assumptions about the extremal dependence type and about the specific structure of statistical models. In this thesis, we develop models with flexible tail dependence structures, in order to provide a reliable estimation of tail characteristics and risk measures. From a methodological perspective, this thesis makes the following novel developments. 1) We propose new copula-based models for multivariate and spatial extremes with flexible tail dependence structures, which are parsimonious and able to bridge smoothly asymptotic dependence and asymptotic independence classes, in both the upper and the lower tails; 2) Moreover, aiming at describing more general dependence structures using graphs, we propose a novel extremal dependence measure called the partial tail-correlation coefficient (PTCC) under the framework of regular variation to learn complex extremal network structures; 3) Finally, we develop a semi-parametric neural-network-based regression model to identify spatial causal effects at all quantile levels (including low and high quantiles). Overall, we make novel contributions to creating new flexible extremal dependence models, developing and implementing novel Bayesian computation algorithms, and taking advantage of machine learning and causal inference principles for modeling extremes. Our novel methodologies are illustrated by a range of applications to financial, climatic, and health data. Specifically, we apply our bivariate copula model to the historical closing prices of five leading cryptocurrencies and estimate the extremal dependence evolution over time, and we use the PTCC to learn the extreme risk network of historical global currency exchange data. Moreover, our multivariate spatial factor copula model is applied to study the upper and lower extremal dependence structures of the daily maximum and minimum air temperature from the state of Alabama in the southeastern United States; and we also apply the PTCC in extreme river discharge network learning for the Upper Danube basin. Finally, we apply the causal spatial quantile regression model in quantifying spatial quantile treatment effects of maternal smoking on extreme low birth weight of newborns in North Carolina, United States.