Tom Henzinger was among the co-founders of the paradigm of hybrid automata in 1992. Hybrid automata possess different locations, holding different ODE-based dynamics; exit conditions from a location trigger transitions, resulting in starting conditions for the next location. A large research activity was developed in the formal verification of hybrid automata; this paradigm still grounds popular commercial tools such as Stateflow for Simulink. However, modeling from first principles of physics requires a different approach: balance equations and conservation laws play a central role, and elementary physical components come with no prespecified input/output profile. All of this leads to grounding physical modeling on DAEs (Differential Algebraic Equations, of the form f (x ′ , x, v) = 0) instead of ODEs. DAE-based modeling, implemented for example in the Modelica language, allows for modularity and reuse of models. Unsurprisingly, DAE-based hybrid systems (also known as multimode DAE systems) emerge as the central paradigm in multiphysics modeling. Despite the growing popularity of modeling tools based on this paradigm, fundamental problems remain in the handling of multiple modes and mode changes-corresponding to multiple locations and transitions in hybrid automata. Deep symbolic analyses (grouped under the term "structural analysis" in the related community), grounded on solid foundations, are required to generate simulation code. This paper reviews the issues related to multimode DAE systems and proposes algorithms for their analysis. Computer science is instrumental in these works, with a lot to offer to the simulation scientific community.