This paper gives a thorough and lucid survey of simultaneous Hermite-Padé approximation to Markov-type functions, i.e., given \(p\) non-negative integers, we have rational approximants with common denominator of the form \[ \pi_{\vec{n}}=\left({Q_{\vec{n},1}\over P_{\vec{n}}},\ldots,{Q_{\vec{n},p}\over P_{\vec{n}}}\right),\;\vec{n}=(n_1,\ldots,n_p), \] to the Markov-functions \[ \vec{f}=(f_1,\ldots,f_p),\;f_j(z)=\int\,{d\mu_j(x)\over z-x}=\sum_{k=0}^{\infty}\,{f_{j,k}\over z^{k+1}} (j=1,\ldots,p), \] with measures \(\mu_j\) supported on the real line and satisfying the degree/order conditions \[ P_{\vec{n}}\not\equiv 0,\;\text{deg}\,P_{\vec{n}}\leq |\vec{n}|=n_1+\cdots +n_p, \] and \[ f_j(z)P_{\vec{n}}(z)-Q_{\vec{n},j}(z)=:R_{\vec{n},j}(z)={\mathcal O}\left({1\over z^{n_j+1}}\right)\text{ as } z\rightarrow\infty, j=1,\ldots, p. \] The denominator satisfies the orthogonality relations \[ \int\,P_{\vec{n}}(x)x^kd\mu_j(x)=0,\;k=0,\ldots,n_j-1,\;j=1,\ldots,p, \] and is referred to as a multiple orthogonal polynomial.
The paper discusses results on convergence and the distributions of poles of the rational approximants and a survey is given of results on the distribution of the eigenvalues of the corrersponding random matrices.
Important tools for the description and proofs are equilibrium potentials and interaction matrices, a notion introduced by Gonchar and Rakhmanov (English translation: [A. A. Gonchar and E. A. Rakhmanov, “On convergence of simultaneous Padé approximants for systems of Markov functions”, Proc. Steklov Inst. Math. 157, 31–50 (198; Zbl 0518.41011) and “Equilibrium measures and the distribution of zeros of extremal polynomials”, Math. USSR, Sb. 53, 119–130 (1986); translation from Mat. Sb., Nov. Ser. 125(167), No. 1, 117–127 (1984; Zbl 0618.30008)].
The layout of the paper is as follows
{Chapter 1: Hermite-Padé approximants} §1: Approximants of Markov-type functions (18 pages) §2: Approximants for functions with complex branch points (11 pages)

{Chapter 2: Multiple orthogonal ensembles} §3: Definitions and determinantal formulae (9 pages) §4: Random matrix model with an external source (10 pages) §5: Non intersecting paths of determinantal point processes (4 pages) §6: The two-matrix model (5 pages)
The paper concludes with a list of 91 references.