We study the problem of evaluating persistent queries over streaming graphs in a principled fashion. These queries need to be evaluated over unbounded and very high speed graph streams. We define a streaming graph data model and query model incorporating navigational queries, subgraph queries and paths as first-class citizens. To support this full-fledged query model we develop a streaming graph algebra that describes the precise semantics of persistent graph queries with their complex constructs. We present transformation rules and describe query formulation and plan generation for persistent graph queries over streaming graphs. Our implementation of a streaming graph query processor shows the feasibility of our approach and allows us to gauge the high performance gains obtained for query processing over streaming graphs.