We completely classify Friedmann-LemaƮtre-Robertson-Walker solutions with spatial curvature K = 0 , \pm 1 and equation of state p = w \rho , according to their conformal structure , singularities and trapping horizons . We do not assume any energy conditions and allow \rho < 0 , thereby going beyond the usual well-known solutions . For each spatial curvature , there is an initial spacelike big-bang singularity for w > -1 / 3 and \rho > 0 , while no big-bang singularity for w < -1 and \rho > 0 . For K = 0 or -1 , -1 < w < -1 / 3 and \rho > 0 , there is an initial null big-bang singularity . For each spatial curvature , there is a final spacelike future big-rip singularity for w < -1 and \rho > 0 , with null geodesics being future complete for -5 / 3 \leq w < -1 but incomplete for w < -5 / 3 . For w = -1 / 3 , the expansion speed is constant . For -1 < w < -1 / 3 and K = 1 , the universe contracts from infinity , then bounces and expands back to infinity . For K = 0 , the past boundary consists of timelike infinity and a regular null hypersurface for -5 / 3 < w < -1 , while it consists of past timelike and past null infinities for w \leq - 5 / 3 . For w < -1 and K = 1 , the spacetime contracts from an initial spacelike past big-rip singularity , then bounces and blows up at a final spacelike future big-rip singularity . For w < -1 and K = -1 , the past boundary consists of a regular null hypersurface . The trapping horizons are timelike , null and spacelike for w \in ( -1 , 1 / 3 ) , w \in \ { 1 / 3 , -1 \ } and w \in ( - \infty, - 1 ) \cup ( 1 / 3 , \infty ) , respectively . A negative energy density ( \rho < 0 ) is possible only for K = -1 . In this case , for w > -1 / 3 , the universe contracts from infinity , then bounces and expands to infinity ; for -1 < w < -1 / 3 , it starts from a big-bang singularity and contracts to a big-crunch singularity ; for w < -1 , it expands from a regular null hypersurface and contracts to another regular null hypersurface .