Stochastic programming is a powerful modeling framework for decision-making under uncertainty. In this work, we tackle two-stage stochastic programs (2SPs), the most widely applied and studied class of stochastic programming models. Solving 2SPs exactly requires evaluation of an expected value function that is computationally intractable. Additionally, having a mixed-integer linear program (MIP) or a nonlinear program (NLP) in the second stage further aggravates the problem difficulty. In such cases, solving them can be prohibitively expensive even if specialized algorithms that exploit problem structure are employed. Finding high-quality (first-stage) solutions – without leveraging problem structure – can be crucial in such settings. We develop Neur2SP, a new method that approximates the expected value function via a neural network to obtain a surrogate model that can be solved more efficiently than the traditional extensive formulation approach. Moreover, Neur2SP makes no assumptions about the problem structure, in particular about the second-stage problem, and can be implemented using an off-the shelf solver and open-source libraries. Our extensive computational experiments on benchmark 2SP datasets from four problem classes with different structures (containing MIP and NLP second-stage problems) show the efficiency (time) and efficacy (solution quality) of Neur2SP. Specifically, the proposed method takes less than 1.66 seconds across all problems, achieving high-quality solutions even as the number of scenarios increases, an ideal property that is difficult to have for traditional 2SP solution techniques. Namely, the most generic baseline method typically requires minutes to hours to find solutions of comparable quality.