The behaviour of semiflexible chains, modelling biopolymers such as DNA and actin in confined spaces, was investigated by means of Monte Carlo simulations. Simulations, based on the coarse-grained worm-like chain (WLC) model, assumed confinement length-scales comparable to those used in micro- and nanofluidic devices. The end-to-end chain elongation R was determined as a function of the channel dimensions and chain bending rigidity. Three regions of chain elongation R, identified in simulations in a cylinder and a slit, were described by current theoretical concepts. In harmony with the measurements of confined DNA, an abrupt transition between the blob region at moderate confinement and the deflection region at strong cylindrical confinement was found. The conditions for hairpin formation were elucidated as a trade-off between confinement and chain stiffness. The intrinsic persistence length of unconfined polymers was calculated by four methods that provided practically identical results. However, in confined geometries only the rigorous and WLC methods predicted the dependence of apparent persistence length P on confinement in a qualitatively correct way. It was found that the simple exponential function, suitable for the description of orientation correlations in free chains is, in confined systems, limited only to short distances along the chain contour and, thus, the apparent persistence length determined by this method just reproduces the intrinsic value of P. The orientation correlations from simulations were compared with analytical predictions in the deflection regime under strong confinement and with the measurements of actin filaments.