This narrative review aimed to elucidate which robot-related characteristics predict relationship formation between typically-developing children and social robots in terms of closeness and trust. Moreover, we wanted to know to what extent relationship formation can be explained by children's experiential and cognitive states during interaction with a robot. We reviewed 86 journal articles and conference proceedings published between 2000 and 2017. In terms of predictors, robots' responsiveness and role, as well as strategic and emotional interaction between robot and child, increased closeness between the child and the robot. Findings about whether robot features predict children's trust in robots were inconsistent. In terms of children's experiential and cognitive states during interaction with a robot, robot characteristics and interaction styles were associated with two experiential states: engagement and enjoyment/liking. The literature hardly addressed the impact of experiential and cognitive states on closeness and trust. Comparisons of children's interactions with robots, adults, and objects showed that robots are perceived as neither animate nor inanimate, and that they are entities with whom children will likely form social relationships. Younger children experienced more enjoyment, were less sensitive to a robot's interaction style, and were more prone to anthropomorphic tendencies and effects than older children. Tailoring a robot's sex to that of a child mainly appealed to boys.