Socratic questioning is a form of reflective inquiry often employed in education to encourage critical thinking in students, and to elicit awareness of beliefs and perspectives in a subject during therapeutic counseling. Specific types of Socratic questions are employed for enabling reasoning and alternate views against the context of individual personal opinions on a topic. Socratic contexts are different from traditional question generation contexts where “answer-seeking” questions are generated against a given formal passage on a topic, narrative stories or conversations. We present SocratiQ, the first large dataset of 110K (question, context) pairs for enabling studies on Socratic Question Generation (SoQG). We provide an in-depth study on the various types of Socratic questions and present models for generating Socratic questions against a given context through prompt tuning. Our automated and human evaluation results demonstrate that our SoQG models can produce realistic, type-sensitive, human-like Socratic questions enabling potential applications in counseling and coaching.