Large-scale sensor deployments and an increased use of privacy-preserving transformations have led to an increasing interest in mining uncertain time series data. Traditional distance measures such as Euclidean distance or dynamic time warping are not always effective for analyzing uncertain time series data. Recently, some measures have been proposed to account for uncertainty in time series data. However, we show in this paper that their applicability is limited. In specific, these approaches do not provide an intuitive way to compare two uncertain time series and do not easily accommodate multiple error functions. In this paper, we provide a theoretical framework that generalizes the notion of similarity between uncertain time series. Secondly, we propose DUST, a novel distance measure that accommodates uncertainty and degenerates to the Euclidean distance when the distance is large compared to the error. We provide an extensive experimental validation of our approach for the following applications: classification, top-k motif search, and top-k nearest-neighbor queries.