Brain networks that govern parental response to infant signals have been studied with imaging techniques over the last 15 years. The complex interaction of thoughts and behaviors required for sensitive parenting enables the formation of each individual's first social bonds and critically shapes development. This review concentrates on magnetic resonance imaging experiments which directly examine the brain systems involved in parental responses to infant cues. First, we introduce themes in the literature on parental brain circuits studied to date. Next, we present a thorough chronological review of state-of-the-art fMRI studies that probe the parental brain with a range of baby audio and visual stimuli. We also highlight the putative role of oxytocin and effects of psychopathology, as well as the most recent work on the paternal brain. Taken together, a new model emerges in which we propose that cortico-limbic networks interact to support parental brain responses to infants. These include circuitry for arousal/salience/motivation/reward, reflexive/instrumental caring, emotion response/regulation and integrative/complex cognitive processing. Maternal sensitivity and the quality of caregiving behavior are likely determined by the responsiveness of these circuits during early parent-infant experiences. The function of these circuits is modifiable by current and early-life experiences, hormonal and other factors. Severe deviation from the range of normal function in these systems is particularly associated with (maternal) mental illnesses - commonly, depression and anxiety, but also schizophrenia and bipolar disorder. Finally, we discuss the limits and extent to which brain imaging may broaden our understanding of the parental brain given our current model. Developments in the understanding of the parental brain may have profound implications for long-term outcomes in families across risk, resilience and possible interventions. This article is part of a Special Issue entitled Oxytocin and Social Behav.