This paper presents a complete analytical characterization of a large class of central and non-central imaging devices dubbed linear cameras by Ponce~\cite{Pon09}. Pajdla~\cite{Pajdla02} has shown that a subset of these, the oblique cameras, can be modelled by a certain type of linear map. We give here a full tabulation of all admissible maps that induce cameras in the general sense of Grossberg and Nayar~\cite{GroNay05}, and show that these cameras are exactly the linear ones. Combining these two models with a new notion of intrinsic parameters and normalized coordinates for linear cameras allows us to give simple analytical formulas for direct and inverse projections. We also show that the epipolar geometry of any two linear cameras can be characterized by a fundamental matrix whose size is at most $6\times 6$ when the cameras are uncalibrated, or by an essential matrix of size at most $4\times 4$ when their internal parameters are known. Similar results hold for trinocular constraints.