Computational models for sarcasm detection have often relied on the content of utterances in isolation. However, the speaker’s sarcastic intent is not always apparent without additional context. Focusing on social media discussions, we investigate three issues: (1) does modeling conversation context help in sarcasm detection? (2) can we identify what part of conversation context triggered the sarcastic reply? and (3) given a sarcastic post that contains multiple sentences, can we identify the specific sentence that is sarcastic? 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 current turn. We show that LSTM networks with sentence-level attention on context and current turn, as well as the conditional LSTM network, outperform the LSTM model that reads only the current turn. As conversation context, we consider the prior turn, the succeeding turn, or both. Our computational models are tested on two types of social media platforms: Twitter and discussion forums. We discuss several differences between these data sets, ranging from their size to the nature of the gold-label annotations. To address the latter two issues, we present a qualitative analysis of the attention weights produced by the LSTM models (with attention) and discuss the results compared with human performance on the two tasks.