With the growing amount of available temporal real-world network data, an important question is how to efficiently study these data. One can simply model a temporal network as either a single aggregate static network, or as a series of time-specific snapshots, each of which is an aggregate static network over the corresponding time window. The advantage of modeling the temporal data in these two ways is that one can use existing well established methods for static network analysis to study the resulting aggregate network(s). Here, we develop a novel approach for studying temporal network data more explicitly. We base our methodology on the well established notion of graphlets (subgraphs), which have been successfully used in numerous contexts in static network research. Here, we take the notion of static graphlets to the next level and develop new theory needed to allow for graphlet-based analysis of temporal networks. Our new notion of dynamic graphlets is quite different than existing approaches for dynamic network analysis that are based on temporal motifs (statistically significant subgraphs). Namely, these approaches suffer from many limitations. For example, they can only deal with subgraph structures of limited complexity. Also, their major drawback is that their results heavily depend on the choice of a null network model that is required to evaluate the significance of a subgraph. However, choosing an appropriate null network model is a non-trivial task. Our dynamic graphlet approach overcomes the limitations of the existing temporal motif-based approaches. At the same time, when we thoroughly evaluate the ability of our new approach to characterize the structure and function of an entire temporal network or of individual nodes, we find that the dynamic graphlet approach outperforms the static graphlet approach, which indicates that accounting for temporal information helps.