We study the shape of the log-returns density $f(x)$ in a CGMY Lévy process $X$ with given skewness $S$ and kurtosis $K$ of $X(1)$ and without a Brownian component. The jump part of such a process is specified by the Lévy density which is $Ce^{-Mx}/x^{1+Y}$ for $x'>0$ and $Ce^{-G|x|}/|x|^{1+Y}$ for $x'<0$. A main finding is that the quantity $R=S^2/K$ plays a major role, and that the class of CGMY processes can be parametrized by the mean, variance, skewness, kurtosis and $Y$, where $Y$ varies in $[0,Y_{\max})$ with $Y_{\max}=(2-3R)/(1-R)$. Limit theorems for $X$ are given in various settings, with particular attention to $X$ approaching Brownian with drift, corresponding to the Black-Scholes model. Implications for moment fitting of log-returns data are discussed. We also exploit the structure of spectrally positive CGMY processes as exponential tiltings (Esscher transforms) of stable processes, with the purpose of providing simple formulas for $f(x)$, short derivations of its asymptotic form, and quick algorithms for simulation and maximum likelihood estimation.