Does a seasonal time series imply a stationary or a non stationary time series If I have a time series that has got seasonality, does that automatically make the series non stationary? My intuition (probably off) is that it does not. 
Seasonality means that the series goes up and down around a constant value....something like a sine wave. So by this logic a time series with seasonality is a (weakly) stationary series (constant mean).
Is this wrong? Why?
 A: IMHO, persistent seasonality, by definition, is a type of non-stationarity: the mean of a seasonal process varies with the season, E[z(t*s+j)] = f(j), where s is the number of seasons, j is a particular season (j=1,...,s), and t is specific period (typically a year). Thus, E[y(t)] = E[sin(t)+u(t)] = sin(t) is not a stable mean, although it is deterministic: you could group observations with different means.
Luis
A: A seasonal pattern that remains stable over time does not make the series non-stationary. A non-stable seasonal pattern, for example a seasonal random walk, will make the data non-stationary.
Edit (after new answer and comments)
A stable seasonal pattern is not stationary in the sense that the mean of the series will vary across seasons and, hence, depends on time; but it is stationary in the sense that we can expect the same mean for the same month in different years.
A stable seasonal pattern may therefore fit in the concept of a cyclostationary process, i.e., a process with a periodic mean and a periodic autocorrelation function.
The above does not apply to a non-stable seasonal pattern.
A: I don`t agree that seasonality is a type of non-stationarity because the concept of stationarity in natural systems already incorporates the idea of fluctuation within an unchanging envelope of variability (Milly et al., 2008).
Speaking about hydrological time-series, even though they are stochastic (random process) and commonly have seasonality, that is, they contain wet and dry periods, they will always be stationary if the mean and the variance do not change over time.
So, ignoring the uncertainties of the effects of climate change, a hydrological time series should normally be stationary even though it still has seasonality.
This is why stationarity is a widely accepted concept for civil engineering design, and because of this concept hydrologists are able to calculate the recurrence time of floods, for example.
Link for Milly et al. (2008): "Stationarity Is Dead: Whither Water Management?"
A: Seasonality does not make your series non-stationary. The stationarity applies to the errors of your data generating process, e.g. $y_t=sin(t)+\varepsilon_t$, where $\varepsilon_t\sim\mathcal{N}(0,\sigma^2)$ and $Cov[\varepsilon_s,\varepsilon_t]=\sigma^21_{s=t}$ is a stationary process, despite having a periodic wave in it, because the errors are stationary.
Seasonality does not make your process stationary either. Consider the same process but $\varepsilon_t\sim\mathcal{N}(0,t\sigma^2)$, in this case the error variance is non-stationary and seasonality has nothing to do with it.
