Most genes are part of larger families of evolutionary-related genes. The history of gene families typically involves duplications and losses of genes as well as horizontal transfers into other organisms. The reconstruction of detailed gene family histories, i.e., the precise dating of evolutionary events relative to phylogenetic tree of the underlying species has remained a challenging topic despite their importance as a basis for detailed investigations into adaptation and functional evolution of individual members of the gene family. The identification of orthologs, moreover, is a particularly important subproblem of the more general setting considered here. In the last few years, an extensive body of mathematical results has appeared that tightly links orthology, a formal notion of best matches among genes, and horizontal gene transfer. The purpose of this chapter is to broadly outline some of the key mathematical insights and to discuss their implication for practical applications. In particular, we focus on tree-free methods, i.e., methods to infer orthology or horizontal gene transfer as well as gene trees, species trees, and reconciliations between them without using a priori knowledge of the underlying trees or statistical models for the inference of phylogenetic trees. Instead, the initial step aims to extract binary relations among genes.