The Costas loop is a modification of the phase-locked loop circuit, which demodulates data and recovers carrier from the input signal. The Costas loop is essentially a nonlinear control system and its nonlinear analysis is a challenging task. Thus, simplified mathematical models and their numerical simulation are widely used for its analysis. At the same time for phase-locked loop circuits there are known various examples where the results of such simplified analysis are differ substantially from the real behavior of the circuit. In this survey the corresponding problems are demonstrated and discussed for the QPSK Costas loop.