To every finite collection of (di-, multi-, hyper-)graphs that is closed under removal of edges one can associate an abstract simplicial complex whose faces are the edge sets of the graphs in the collection. Graph complexes have provided an important link between combinatorics and algebra, topology and geometry. The author considers the simplicial complexes associated with the subgraph collection of a graph \(G\) whose node degrees are bounded from above. Some special cases are the matching complex (\(G\) is the complete graph and degree bounds are \(1\)) and the chessboard complex (\(G\) is the complete bipartite graph and the degree bounds are 1). The author discusses the earliest results in this area, dating back to the 1970s, as well as the most recent developments. Many of these results are quite elegant and their proofs make use of powerful combinatorial techniques from computing homotopy type, homology and representations of the symmetric group on homology.