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This paper shows how two of the most important analysis methods for Petri nets can be performed in a modular way. We illustrate our techniques by means of modular Place/Transitions nets (modular PT-nets) in which the individual modules interact via shared places and shared transitions. For place invariants we show that it is possible to construct invariants of the total modular PT-net from invariants of the individual modules. For state spaces, we show that it is possible to decide behavioural properties of the modular PT-net from state spaces of the individual modules plus a synchronization graph, without unfolding to the ordinary state space. The generalization of our techniques to high-level Petri nets is rather straightforward.