We introduce the class of interval k-graphs, which is the generalization of interval graphs, particularly interval bigraphs. For a fixed k, we say that an input graph G with given partition \(V_1, \dots , V_k\) of its vertices into independent sets is an interval k-graph if each vertex \(v \in G\) can be represented by an interval \(I_v\) from a real line so that \(u \in V_i\) and \(v \in V_j\) , \(i \ne j\) are adjacent if and only if the intervals \(I_u\) and \(I_v\) intersect. We study the ordering characterization and forbidden obstructions of interval k-graphs and present a polynomial-time recognition algorithm for them. Special cases of interval k-graphs, particularly comparability interval k-graphs, were previously studied in [2], where the complexity of interval k-graph recognition was posed as an open problem.