Many important underlying concepts and analytic features of dynamic linear models are apparent in the simplest and most widely used case of the *first-order polynomial model*. By way of introduction to DLMs, this case is described and examined in detail in this Chapter. The first-order polynomial model is the simple, yet non-trivial, time series model in which the observation series *Y*_{t} is represented as *Y*_{t} = μ_{t} + ν_{t}, μ_{t} being the current *level* of the series at time *t*, and ν_{t} ∼ N[0, *V*_{t}] the *observational error* or noise term. The *time evolution* of the level of the series is a simple random walk μ_{t} = μ_{t−1} + ω_{t}, with *evolution error* ω_{t} ∼ N[0, *W*_{t}]. This latter equation describes what is often referred to as a *locally constant mean model*. Note the assumption that the two error terms, observational and evolution errors, are normally distributed for each *t*. In addition we adopt the assumptions that the error sequences are independent over time and mutually independent. Thus, for all *t* and all *s* with *t* ≠ *s*, ε_{t} and ε_{s} are independent, ω_{t} and ω_{s} are independent, and ν_{t} and ω_{s} are independent. Further assumptions at this stage are that the variances *V*_{t} and *W*_{t} are known for each time *t*. Figure 2.1 shows two examples of such *Y*_{t} series together with their underlying μ_{t} processes. In each the starting value is μ_{0} = 25, and the variances defining the model are constant in time, *V*_{t} = *V* and *W*_{t} = *W*, having values *V* = 1 in both cases and evolution variances (a) *W* = 0.05, (b) *W* = 0.5. Thus in (a) the movement in the level over time is small compared to the observational variance, *W* = *V*/20, leading to a typical locally constant realisation, whereas in (b) the larger value of *W* leads to greater variation over time in the level of the series.