Capsule　networks　have　drawn　increasing　attention　since　the　concept　is　proposed　in　2011.　They　hold　advantages　over　convolutional　neural　networks　(CNNs)　on　modeling　part-To-whole　relationships　between　entities　and　learning　viewpoint　invariant　representations.　A　variety　of　improved　designs　and　applications　related　to　capsule　networks　have　been　explored　within　the　past　decade.　This　paper　presents　a　relatively　thorough　survey　of　literatures　corresponding　to　the　evolution　of　the　capsule　networks　and　their　applications.　Different　design　of　capsule　networks　from　Hinton＇s　group　as　well　as　other　researchers　are　reviewed.　Applications　of　capsule　networks　in　various　scenarios　and　tasks　are　elaborated,　including　hyperspectral　image　processing,　medical　image　processing,　facial　image　processing,　text　classification,　object　detection,　image　segmentation,　few-shot　learning,　etc.　The　relation　and　comparison　of　capsule　networks,　CNNs,　and　transformers,　and　the　trend　of　future　development　of　capsule　networks　are　discussed.　We　believe　this　review　provides　a　good　reference　to　researchers　who　are　interested　in　capsule　networks.　©　2021　IEEE.