Host-cell invasion by Cryptosporidium is a complex process that requires many different factors derived from both the parasite and the host cell. However, the exact natures of the processes have yet to be resolved. Here, research on different components of the invasion process is put in context, and the sequence of events and pathways associated with the establishment of Cryptosporidium in its unique niche is clarified. In addition, initial parasite–host contact, host-cell invasion and host-cell responses are described. The roles of parasite and host-cell-derived components in the invasion process are examined, as is the question of whether Cryptosporidium actively invades cells and to what extent host-cell responses are involved.