This chapter presents an analytical study of the Bees Algorithm, aiming to define the algorithm and analyse its behaviour mathematically. It begins by providing a formal definition of the Bees Algorithm, including its components and parameters. The chapter then focuses on analysing the local search behaviour, particularly the concept of “reach,” which represents the extent of the search area around a solution. It derives bounds on the reach and explores the expected progress of the local search. The chapter also examines the phenomenon of local search stalling, where the search fails to find better solutions within the neighbourhood. It introduces the concept of site abandonment, which helps prevent the algorithm from getting stuck in unproductive areas. The probability of stalling is analysed with and without neighbourhood shrinking, a technique that reduces the search area over time. Furthermore, the chapter investigates the impact of neighbourhood shape on the search process, considering both isotropic and non-isotropic local search. It also analyses the global search component of the Bees Algorithm, which involves exploring the solution space randomly. The chapter concludes by discussing the insights gained from the analytical study and summarising the key findings.