Children, adolescents, and young adults with at least one first-degree relative [familial high-risk (FHR)] with either schizophrenia (SZ) or bipolar disorder (BD) have a one-in-two risk of developing a psychiatric disorder. Here, we review functional magnetic resonance imaging (fMRI) studies which examined task-related brain activity in young individuals with FHR-SZ and FHR-BD. A systematic search identified all published task-related fMRI studies in children, adolescents, and young adults below an age of 27 years with a first-degree relative with SZ or BD, but without manifest psychotic or affective spectrum disorder themselves. The search identified 19 cross-sectional fMRI studies covering four main cognitive domains: 1) working memory (n = 3), 2) cognitive control (n = 4), 3) reward processing (n = 3), and 4) emotion processing (n = 9). Thirteen studies included FHR-BD, five studies included FHR-SZ, and one study included a pooled FHR group. In general, task performance did not differ between the respective FHR groups and healthy controls, but 18 out of the 19 fMRI studies revealed regional alterations in task-related activation. Brain regions showing group differences in peak activation were regions associated with the respective task domain and showed little overlap between FHR-SZ and FHR-BD. The low number of studies, together with the low number of subjects, and the substantial heterogeneity of employed methodological approaches within the domain of working memory, cognitive control, and reward processing impedes finite conclusions. Emotion processing was the most investigated task domain in FHR-BD. Four studies reported differences in activation of the amygdala, and two studies reported differences in activation of inferior frontal/middle gyrus. Together, these studies provide evidence for altered brain processing of emotions in children, adolescents, and young adults at FHR-BD. More studies of higher homogeneity, larger sample sizes and with a longitudinal study design are warranted to prove a shared or specific FHR-related endophenotypic brain activation in young first-degree relatives of individuals with SZ or BD, as well as to pinpoint specific alterations in brain activation during cognitive-, emotional-, and reward-related tasks.