Molecular evolutionary studies provide a means of investigating how cells function and how organisms adapt to their environment. The products of evolutionary studies provide medically important insights to the source of major diseases, such as HIV, and hold the key to understand the developing immunity of pathogenic bacteria to antibiotics. They have also helped mankind understand its place in nature, casting light on the selective forces and environmental conditions that resulted in modern humans. The use of likelihood as a framework for statistical modeling in phylogenetics has played a fundamental role in studying molecular evolution, enabling rigorous and robust conclusions to be drawn from sequence data. The first half of this article is a general introduction to the likelihood method for inferring phylogenies, the properties of the models used, and how it can be used for statistical testing. The latter half of the article focuses on the emerging new generation of phylogenetic models that describe heterogeneity in the evolutionary process along sequences, including the recoding of protein coding sequence data to amino acids and codons, and various approaches for describing dependencies between sites in a sequence. We conclude with a detailed case study examining how modern modeling approaches have been successfully employed to identify adaptive evolution in proteins.