Cracks and cavities belong to two basic forms of damage to the concrete structure, which may reduce the load-bearing capacity and tightness of the structure and lead to failures and catastrophes in construction structures. Excessive and uncontrolled cracking of the structural element may cause both corrosion and weakening of the adhesion of the reinforcement present in it. Moreover, cracking in the structure negatively affects its aesthetics and in extreme cases may cause discomfort to people staying in such a building. Therefore, the following article provides an in-depth review of issues related to the formation and development of damage and cracking in the structure of concrete composites. It focuses on the causes of crack initiation and characterizes their basic types. An overview of the most commonly used methods for detecting and analyzing the shape of microcracks and diagnosing the trajectory of their propagation is also presented. The types of cracks occurring in concrete composites can be divided according to eight specific criteria. In reinforced concrete elements, macrocracks depend on the type of prevailing loads, whereas microcracks are correlated with their specific case. The analyses conducted show that microcracks are usually rectilinear in shape in tensioned elements; in shear elements there are wing microcracks with straight wings; and torsional stresses cause changes in wing microcrack morphology in that the tips of the wings are twisted. It should be noted that the subject matter of microcracks and cracks in concrete and structures made of this material is important in many respects as it concerns, in a holistic approach, the durability of buildings, the safety of people staying in the buildings, and costs related to possible repairs to damaged structural elements. Therefore, this problem should be further investigated in the field of evaluation of the cracking and fracture processes, both in concrete composites and reinforced concrete structures.