Over the past few decades of development， microphone array technology is becoming more mature， which has been applied to various human-machine interaction systems， e.g.， video-conferencing， intelligent television， mobile telephony， hearing aids. However， in realistic noisy or distant interaction scenarios， the sound acquisition quality （SAQ） of conventional topology-constrained microphone arrays cannot be guaranteed. With the wide range of using wireless intelligent terminal devices， distributed microphone array （DMA） or so-called wireless acoustic sensor network （WASN） provides more possibilities of improving the SAQ for speech interaction systems in complex and open domains， and shows a superiority in array organization， application experience and scene coverage. Recently， DMA exhibits a good applicable potential in many speech interaction tasks， which almost cover the tasks that conventional microphone array can handle. This survey will mainly summarize some existing important sound acquisition theories and application methods of DMA， including principles of array organization， utility evaluation of microphone nodes and the application methods in combination of downstream speech tasks. Finally， we will briefly discuss some key challenges and developing trends of the road of DMA to practical usages.