The aim of this article is to provide several ways to assess the observability and the reconstructibility of bounded cellular automata (CA). First, generic definitions for the considered CA and their associated sensors are given, as well as the definitions for observability and reconstructibility of these measured dynamical systems. Criteria of observability and reconstructibility are proposed for linear and affine CA through an extension of the Kalman criterion and for non-linear CA from an extension of the results on observability and reconstructibility of Boolean networks. A method for decentralised observability calculation is presented for non-linear CA in order to reduce the algorithmic complexity. These criteria are analyzed and illustrated through several examples, either with fixed or mobile sensors. Numerical methods and related complexity issues are discussed.