Large-scale databases of human activity in social media have captured scientific and policy attention, producing a flood of research and discussion. This paper considers methodological and conceptual challenges for this emergent field, with special attention to the validity and representativeness of social media big data analyses. Persistent issues include the over-emphasis of a single platform, Twitter, sampling biases arising from selection by hashtags, and vague and unrepresentative sampling frames. The socio-cultural complexity of user behavior aimed at algorithmic invisibility (such as subtweeting, mock-retweeting, use of "screen captures" for text, etc.) further complicate interpretation of big data social media. Other challenges include accounting for field effects, i.e. broadly consequential events that do not diffuse only through the network under study but affect the whole society. The application of network methods from other fields to the study of human social activity may not always be appropriate. The paper concludes with a call to action on practical steps to improve our analytic capacity in this promising, rapidly-growing field.