Extremal and approximative properties of simple partial fractions

In approximation theory, logarithmic derivatives of complex polynomials are called simple partial fractions (SPF) as suggested by E.P. Dolzhenko. Many solved and unsolved extremal problems related to SPF are traced back to works of G. Boole, A.J. Macintyre, W.H.J. Fuchs, J.M. Marstrand, E.A. Gorin, A.A. Gonchar, E.P. Dolzhenko. At present, many authors systematically develop methods for approximation and interpolation by SPF and several their modifications. Simultaneously, related problems, being of independent interest, arise for SPF: inequalities of different metrics, estimation of derivatives, separation of singularities, etc.
We systematize some of these problems which are known to us in Introduction of this survey. In the main part, we formulate principal results and outline methods to prove them if possible.