We consider the problems of estimation, learning, and optimization
over  a large dataset of which a subset consists of points drawn from
the distribution of interest, and make no assumptions on the remaining
points---they could be well-behaved, extremely biased, or
adversarially generated.  We investigate this question via the
introduction of two new model for studying robust estimation,
learning, and optimization.  One of these models, which we term the
"semi-verified" model, assumes access to a second much smaller
(typically constant-sized) set of "verified" or trusted data points
that have been drawn from the distribution of interest.   The key
question in this model is how a tiny, but trusted dataset can allow
for the accurate extraction of the information contained in the large,
but untrusted dataset.  The second model we propose, which we term
"list-decodable learning", considers the task of returning a small
list of proposed answers.  Underlying this model is the question of
whether the presence of a significant amount of adversarial datapoints
can mask the structure of the "good" datapoints, or whether it can
simply create additional false copies of the structure of the good
points.