Let $\mathcal F$ be a general separable metric space and denote by $\mathcal D_n=\{(\bX_1,Y_1), \hdots, (\bX_n,Y_n)\}$ independent and identically distributed $\mathcal F\times \mathbb R$-valued random variables with the same distribution as a generic pair $(\bX, Y)$. In the regression function estimation problem, the goal is to estimate, for fixed $\bx \in \mathcal F$, the regression function $r(\bx)=\mathbb E[Y|\bX=\bx]$ using the data $\mathcal D_n$. Motivated by a broad range of potential applications, we propose, in the present contribution, to investigate the properties of the so-called $k_n$-nearest neighbor regression estimate. We present explicit general finite sample upper bounds, and particularize our results to important function spaces, such as reproducing kernel Hilbert spaces, Sobolev spaces or Besov spaces.