Summary: Longevity risk constitutes an important risk factor for insurance companies and pension plans. For its analysis, but also for evaluating mortality-contingent structured financial products, modeling approaches allowing for uncertainties in mortality projections are needed. One model class that has attracted interest in applied research as well as among practitioners are forward mortality models, which are defined based on forecasts of survival probabilities as can be found in generation life tables and infer dynamics on the entire age/term structure, or forward surface, of mortality. However, thus far, there has been little guidance on identifying suitable specifications and their properties. The current paper provides a detailed analysis of forward mortality models driven by a finite-dimensional Brownian motion. In particular, after discussing basic properties, we present an infinite-dimensional formulation, and we examine the existence of finite-dimensional realizations for time-homogeneous deterministic volatility models, which are shown to possess important advantages for practical applications.