Computational models for sarcasm detection have often relied on the content of utterances in isolation. However, speaker’s sarcastic intent is not always obvious without additional context. Focusing on social media discussions, we investigate two issues: (1) does modeling of conversation context help in sarcasm detection and (2) can we understand what part of conversation context triggered the sarcastic reply. To address the first issue, we investigate several types of Long Short-Term Memory (LSTM) networks that can model both the conversation context and the sarcastic response. We show that the conditional LSTM network (Rocktäschel et al. 2015) and LSTM networks with sentence level attention on context and response outperform the LSTM model that reads only the response. To address the second issue, we present a qualitative analysis of attention weights produced by the LSTM models with attention and discuss the results compared with human performance on the task.