We discuss a new class of non-spherical cage carbon clusters - barrelenes - which manifest themselves in a stable or metastable solid phase. They present a cylindrical graphite surface with an elemental frame-belt of 2m (m = 5, 6, 7,...) carbon atoms and two spherical-like appropriate lids. As an example, the barrelenes C24+12n (n is the number of belts, m = 6 is the number of carbon hexagons in a belt) and C60+18n (m = 9) are analyzed in detail. The possible types of lids are described: closed - bowl-shaped, partially open - collar-shaped, and corresponding clusters: (a) b-C(N) barrelenes - barrel-shaped (n < m) clusters with closed lids; (b) j-C(N) jarenes - jar-shaped n < m) clusters with one partially open lid; (c) m-C(N) muffenes - muff-shaped (n < m) clusters with both lids open; (d) t-C(N) tubelenes - tube-like (n greater-than-or-equal-to) clusters with any kind of lids. The main Kekule structures of such clusters are analyzed. Possible b-C(N) and t-C(N) solids and their doped analogues are discussed including the properties of an isolated long (n >> m) tubelene which is a quasi-one-dimensional solid. The physical properties of such solids (conduction anisotropy, local piezoelectric-like effect, optic non-linearity, possible superconductivity) are considered.