Rapid access to situation-sensitive data through social media networks creates new opportunities to address a number of real-world problems. Damage assessment during disasters is a core situational awareness task for many humanitarian organizations that traditionally takes weeks and months. In this work, we analyze images posted on social media platforms during natural disasters to determine the level of damage caused by the disasters. We employ state-of-the-art machine learning techniques to perform an extensive experimentation of damage assessment using images from four major natural disasters. We show that the domain-specific fine-tuning of deep Convolutional Neural Networks (CNN) outperforms other state-of-the-art techniques such as Bag-of-Visual-Words (BoVW). High classification accuracy under both event-specific and cross-event test settings demonstrate that the proposed approach can effectively adapt deep-CNN features to identify the severity of destruction from social media images taken after a disaster strikes.