The molecular-replacement method works well with good models and simple unit cells, but often fails with more difficult problems. Experience with likelihood in other areas of crystallography suggests that it would improve performance significantly. For molecular replacement, the form of the required likelihood function depends on whether there is ambiguity in the relative phases of the contributions from symmetry-related molecules (e.g. rotation versus translation searches). Likelihood functions used in structure refinement are appropriate only for translation (or six-dimensional) searches, where the correct translation will place all of the atoms in the model approximately correctly. A new likelihood function that allows for unknown relative phases is suitable for rotation searches. It is shown that correlations between sequence identity and coordinate error can be used to calibrate parameters for model quality in the likelihood functions. Multiple models of a molecule can be combined in a statistically valid way by setting up the joint probability distribution of the true and model structure factors as a multivariate complex normal distribution, from which the conditional distribution of the true structure factor given the models can be derived. Tests in a new molecular-replacement program, Beast, show that the likelihood-based targets are more sensitive and more accurate than previous targets. The new multiple-model likelihood function has a dramatic impact on success.