Most have beautiful pleated foliage that makes them especially useful for bonsai, although they also make good ornamental trees. All require protection from afternoon sun in hottest driest locations.
Hornbeams make exceptional group plantings as well as specimen bonsai.
The Asian species are traditional bonsai species, but the American hornbeam, C. caroliana , will make equally good bonsai. The growth rate of the hornbeam species varies greatly. This is of particular importance if you are growing them out for larger trunked bonsai. C. coreana is very popular because of its small leaves and very ramified growth pattern. However, it takes a signifcantly longer time to achieve trunk caliper.