Chemically Induced Dynamic Nuclear Polarization (CIDNP) is an efficient method of creating non-equilibrium polarization of nuclear spins by using chemical reactions, which have radical pairs as intermediates. The CIDNP effect originates from (i) electron spin-selective recombination of radical pairs and (ii) the dependence of the inter-system crossing rate in radical pairs on the state of magnetic nuclei. The CIDNP effect can be investigated by using Nuclear Magnetic Resonance (NMR) methods. The gain from CIDNP is then two-fold: it allows one to obtain considerable amplification of NMR signals; in addition, it provides a very useful tool for investigating elusive radicals and radical pairs. While the mechanisms of the CIDNP effect in liquids are well established and understood, detailed analysis of solid-state CIDNP mechanisms still remains challenging; likewise a common theoretical frame for the description of CIDNP in both solids and liquids is missing. Difficulties in understanding the spin dynamics that lead to the CIDNP effect in the solid-state case are caused by the anisotropy of spin interactions, which increase the complexity of spin evolution. In this work, we propose to analyze CIDNP in terms of level crossing phenomena, namely, to attribute features in the CIDNP magnetic field dependence to Level Crossings (LCs) and Level Anti-Crossings (LACs) in a radical pair. This approach allows one to describe liquid-state CIDNP; the same holds for the solid-state case where anisotropic interactions play a significant role in CIDNP formation. In solids, features arise predominantly from LACs, since in most cases anisotropic couplings result in perturbations, which turn LCs into LACs. We have interpreted the CIDNP mechanisms in terms of the LC/LAC concept. This consideration allows one to find analytical expressions for a wide magnetic field range, where several different mechanisms are operative; furthermore, the LAC description gives a way to determine CIDNP sign rules. Thus, LCs/LACs provide a consistent description of CIDNP in both liquids and solids with the prospect of exploiting it for the analysis of short-lived radicals and for optimizing the polarization level.