Prospective memory (PM) is the ability to remember and realize one’s intentions in the future; therefore, it is crucial for the daily functioning of children and adolescents and their ability to become independent from caregivers. PM errors can have repercussions during childhood, such as influencing school performance and social relationships. The aim of this systematic review was to synthesize studies analysing PM in children and adolescents (age range: 0–16 years) following PRISMA guidelines. The goal was to outline the most commonly used tasks, offering information on the development of PM, and—through a detailed analysis of the assessment of specific cognitive processes carried out in the primary studies included—providing information on the main cognitive processes involved in PM within this age group. Forty-nine studies were selected that examined PM in children and adolescents with typical development. The studies used many different tasks that can be traced back to eleven different main paradigms to evaluate PM, each structured into a PM and an ongoing task. Older children performed better on PM targets than younger children, suggesting a developmental trajectory of PM that follows a J-shaped function. Children as young as 2 years old exhibited the first signs of PM, while adolescents performed similarly to adults on PM tasks. Several factors are involved in PM development: retrospective memory, executive functions (planning, working memory, inhibitory control, monitoring), attention, metamemory, and motivation. This review May be considered a starting point to summarize the most used tools to evaluate PM in children and adolescents, and to shed light on the primary cognitive functions involved in PM, potentially offering indications to researchers in selecting optimal tasks for measuring PM across different age groups. Additionally, it underscores the importance of developing standardized measures for potential clinical applications.