Any filtered complex over a field F can be brought by a linear transformation preserving the filtration to so called canonical form, a canonically defined direct sum of filtered complexes of two types: one-dimensional complexes with trivial differential $d(e_{t_i})=0$ and two-dimensional complexes with trivial homology $d(e_{s_j})=e_{r_j}$. The tutorial is devoted to the proof of this theorem, that was first published in the speaker’s 1994 paper “Framed Morse complex and its invariants”, AMS, Advances in Soviet Mathematics, 21: 93–115, and reported at the 1994 International Congress of Mathematicians satellite conference in Konstanz (Germany). Starting from the early 2000s these invariants became widely popular in Applied Mathematics under the name of “persistence diagrams” and “persistence barcodes”. The mentioned classification theorem is usually referred to in Applied Mathematics as the Persistent homology Main (or Structure, or Principal) Theorem. More than 15 different software platforms, that exist actually for computation of persistence diagrams, are based on the algorithm, described in the mentioned 1994 paper, that brings filtered complexes to the canonical form.