CPsuperH: a computational tool for Higgs phenomenology in the minimal supersymmetric standard model with explicit CP-violation. We provide a detailed description of the Fortran code CPsuperH, a newly-developed computational package that calculates the mass spectrum and decay widths of the neutral and charged Higgs bosons in the Minimal Supersymmetric Standard Model with explicit CP violation. The program is based on recent renormalization-group-improved diagrammatic calculations that include dominant higher-order logarithmic and threshold corrections, b-quark Yukawa-coupling resummation effects and Higgs-boson pole-mass shifts. The code CPsuperH is self-contained (with all subroutines included), is easy and fast to run, and is organized to allow further theoretical developments to be easily implemented. 1 The fact that the masses and couplings of the charged and neutral Higgs bosons are computed at a similar high-precision level makes it an attractive tool for Tevatron, LHC and LC studies, also in the CP-conserving case.