The channel capacity of near-field (NF) communications is characterized by considering three types of line-of-sight multiuser channels: i) multiple access channel (MAC), ii) broadcast channel (BC), and iii) multicast channel (MC). For NF MAC and BC, closed-form expressions are derived for the sum-rate capacity as well as the capacity region under a two-user scenario. These results are further extended to scenarios with an arbitrary number of users. For NF MC, closed-form expressions are derived for the two-user channel capacity and the capacity upper bound with more users. Further insights are gleaned by exploring special cases, including scenarios with infinitely large array apertures, co-directional users, and linear arrays. For comparison, the MAC and BC sum-rates achieved by typical linear combiners and precoders are also analyzed. Theoretical and numerical results are presented and compared with far-field communications to demonstrate that: i) the NF capacity of these three channels converges to finite values rather than growing unboundedly as the number of array elements increases; ii) the capacity of the MAC and BC with co-directional users can be improved by using the additional range dimensions in NF channels to reduce inter-user interference (IUI); and iii) the MC capacity benefits less from the NF effect compared to the MAC and BC, as multicasting is less sensitive to IUI.