The authors propose models that can fit the observed prices of liquid instruments in a similar fashion to the market models, but which also have the advantage that prices can be calculated just as efficiently as in the short-rate models of the traditional approach. To achieve this, it is considered the general class of Markov-functional interest rate models, the defining characteristic of which is that pure discount bond prices are at any time a function of some low-dimensional process which is Markovian in some martingale measure. This ensures that implementation is efficient since it is only necessary to track the driving Markov process. Market models do not possess this property (for a low-dimensional Markov process) and this is the impediment to their efficient implementation. The freedom to choose the functional form is what permits accurate calibration of Markov-functional models to relevant market prices, a property not possessed by short-rate models. The remaining freedom to specify the law of the driving Markov process is what allows to make the model realistic. The focus of the most papers is primarily on the form of the Markov process. By contrast, the focus of this paper is on the functional forms and showing how they can be chosen such that the Markov-functional model has certain pre-specified properties. In particular, it is shown how to generate Markov-functional models that replicate market prices for LIBOR- and swap-based instruments.