We present a neural network architecture based on bidirectional LSTMs to compute representations of words in the sentential contexts. These context-sensitive word representations are suitable for, e.g., distinguishing different word senses and other context-modulated variations in meaning. To learn the parameters of our model, we use cross-lingual supervision, hypothesizing that a good representation of a word in context will be one that is sufficient for selecting the correct translation into a second language. We evaluate the quality of our representations as features in three downstream tasks: prediction of semantic supersenses (which assign nouns and verbs into a few dozen semantic classes), low resource machine translation, and a lexical substitution task, and obtain state-of-the-art results on all of these.