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We elaborate on conformal higher-spin gauge theory in three-dimensional (3D) curved space. For any integer n > 2 we introduce a conformal spin-n2 gauge field h(n)=hα1…αn (with n spinor indices) of dimension (2 − n/2) and argue that it possesses a Weyl primary descendant C(n) of dimension (1 + n/2). The latter proves to be divergenceless and gauge invariant in any conformally flat space. Primary fields C(3) and C(4) coincide with the linearised Cottino and Cotton tensors, respectively. Associated with C(n) is a Chern-Simons-type action that is both Weyl and gauge invariant in any conformally flat space. These actions, which for n = 3 and n = 4 coincide with the linearised actions for conformal gravitino and conformal gravity, respectively, are used to construct gauge-invariant models for massive higher-spin fields in Minkowski and anti-de Sitter space. In the former case, the higher-derivative equations of motion are shown to be equivalent to those first-order equations which describe the irreducible unitary massive spin-n2 representations of the 3D Poincaré group. Finally, we develop N= 1 supersymmetric extensions of the above results. © 2018, The Author(s).