There has been a lot of effort to construct good quantum codes from the classical error correcting codes. Constructing new quantum codes, using Hermitian self-orthogonal codes, seems to be a difficult problem in general. In this paper, Hermitian self-orthogonal codes are studied from algebraic function fields. Sufficient conditions for the Hermitian self-orthogonality of an algebraic geometry code are presented. New Hermitian self-orthogonal codes are constructed from projective lines, elliptic curves, hyper-elliptic curves, Hermitian curves, and Artin-Schreier curves. In addition, over the projective lines, we construct new families of MDS quantum codes with parameters $[[N,N-2K,K+1]]_q$ under the following conditions: i) $N=t(q-1)+1$ or $t(q-1)+2$ with $t|(q+1)$ and $K=\lfloor\frac{t(q-1)+1}{2t}\rfloor+1$; ii) $(n-1)|(q^2-1)$, $N=n$ or $N=n+1$, $K_0=\lfloor\frac{n+q-1}{q+1}\rfloor$, and $K\ge K_0+1$; iii) $N=tq+1$, $\forall~1\le t\le q$ and $K=\lfloor\frac{tq+q-1}{q+1}\rfloor+1$; iv) $n|(q^2-1)$, $n_2=\frac{n}{\gcd (n,q+1)}$, $\forall~ 1\le t\le \frac{q-1}{n_2}-1$, $N=(t+1)n+2$ and $K=\lfloor \frac{(t+1)n+1+q-1}{q+1}\rfloor+1$.