Suppose $(X_1,Y_1),(X_2,Y_2),\ldots,(X_n,Y_n)$ are i.i.d random vectors with a continuous distribution. Let $R_i =\operatorname{Rank}(X_i)$ among $X_1,X_2,\ldots,X_n$ and $Q_i=\operatorname{Rank}(Y_i)$ among $Y_1,Y_2,\ldots,Y_n$, $\,i=1,2,\ldots,n$. 

Spearman's rank correlation coefficient is then the sample quantity

$$r_S=1-\frac{6}{n(n^2-1)}\sum_{i=1}^n (R_i-Q_i)^2$$

It is not difficult to show (for example, using the theory of U-statistics) that as $n\to \infty$, $$E(r_S)\to \rho_G\,,$$ 

where $\rho_G$ is the *grade correlation* defined as

$$\rho_G=\operatorname{Corr}(F(X_1),G(Y_1))$$

Here $F$ and $G$ are the distribution functions of $X$ and $Y$ respectively. 

So $r_S$ is an asymptotically unbiased estimator of $\rho_G$, and in this sense $\rho_G$ is a reasonable parameter of interest. 

Related question: https://stats.stackexchange.com/q/161695/119261.