We introduce team pushdown automata as a theoretical framework capable of modelling various communication and cooperation strategies in complex, distributed systems. Team pushdown automata are obtained by augmenting distributed pushdown automata with the notion of team cooperation or - alternatively - by augmenting team automata with pushdown memory. In a team pushdown automaton, several pushdown automata work as a team on the input word placed on a common one-way input tape. At any moment in time one team of pushdown automata, each with the same symbol on top of its stack, is active: each pushdown automaton in the active team replaces the topmost symbol of its stack and changes state, while the current input symbol is read from the input tape by a common reading head. The teams are formed according to the team cooperation strategy of the team pushdown automaton and may vary from one moment to the other. Based on the notion of competence, we introduce a variety of team cooperation strategies. If all stacks are empty when the input word has been completely read, then this word is part of the language accepted by the team pushdown automaton. Here we focus on the accepting capacity of team pushdown automata.