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Over the past four decades, two distinct likelihood-based estimation methods for Exponential-family Random Graph Models (ERGMs) have emerged. One method tries, due to the complexity of the model, to approximate the maximum likelihood estimator (MLE) using Markov Chain Monte Carlo (MCMC) techniques, while the other method optimizes a simpler, but misspecified likelihood function that results in the so-called maximum pseudolikelihood estimator (MPLE). Interestingly, both estimators can be seen as opposite ends of a spectrum in multiple senses. In this dissertation, I examine the theory of these two estimation approaches and, in the process, develop an improved method for determining starting values for the MCMC algorithm that is based on the likelihood principle and uses simulated annealing. I also introduce two approaches for correctly approximating standard errors for the MPLE. These two approaches are based on parametric bootstrapping and the Godambe information matrix, respectively. This dissertation also provides empirical evidence that it is possible, by using an offset term that depends on the sample size, to embed some models in a sequence that renders the MPLE consistent and asymptotically normal. I focus the applications of the proposed techniques on networks predominantly explored in the field of political science. In a first project, a citation ERGM is developed to analyze the citation network among US Supreme Court majority opinions. In addition to finding evidence of exogenous covariates, the model provides evidence for reciprocity, transitivity, and popularity in the network. The other project examines the network of bills introduced in the United State Senate, where ties between bills indicate the similarity of the set of legislators who co-sponsor them.