We present a family of static and evolving spherically symmetric Lorentzian wormhole solutions in N + 1-dimensional Einstein gravity. In general, for static wormholes, we require that, at least, the radial pressure has a barotropic equation of state of the form p(r) = omega(r)rho, where the state parameter omega(r) is constant. On the other hand, it is shown that, in any dimension N >= 3, with phi(r) = Lambda = 0 and anisotropic barotropic pressure with constant state parameters, static wormhole configurations are always asymptotically flat spacetimes, while, in 2 + 1 gravity, there are not only asymptotically flat static wormholes but also more general ones. In this case, the matter sustaining the three-dimensional wormhole may be only a pressureless fluid. In the case of evolving wormholes with N >= 3, the presence of a cosmological constant leads to an expansion or contraction of the wormhole configurations: for positive cosmological constants, we have wormholes which expand forever, and, for negative cosmological constants, we have wormholes which expand to a maximum value and then recollapse. In the absence of a cosmological constant, the wormhole expands with constant velocity, i.e., without acceleration or deceleration. In 2 + 1 dimensions, the expanding wormholes always have an isotropic and homogeneous pressure, depending only on the time coordinate.