Modern age pure bred dog breeding has a lot of facets and many problems. Let me just single out a few of them for discussion since those are the most pressing to me.
1. Health problems due to small gene pools
Population genetics tells us the what we need to know to breed healthy predictable dog types (breeds) in a sustainable way. Unfortunately for the last 150 years pure bred dog breeding has followed a different path. Modern dog breeding started in the Victorian age, when the notion of nobility was important. Therefore dogs had to be noble, too, descending from noble ancestors and no minor blood should dilute the noble blood. The idea of "breed" was no longer defined by looks or abilities but by descendence. A breed was developed and a number of founder dogs were used and registered in the stud book. As soon as satisfying type was achieved, the stud book was closed and henceforth only dogs descending from the original founders could claim to be of this noble breed.
It is a logical fact that the gene pool i.e. the number of available genetic variants can never be bigger as the number of variants those founder dogs have contributed. In each subsequent generation there is an inevitable loss of alleles as not all dogs are bred. With every dog culled from the breeding population for whatever reason all the alleles carried in its genome are lost. To fix a type and get predictability and uniformity breeding related dogs to each other (line breeding, inbreeding) was the favoured technique. It guaranteed fast and easy success in terms of fixing traits in the population. A trait gets fixed better if there are no alternative alleles available so homozygousity was aimed for.
The breeders of old did not know that every individual, be it human, dog, mouse or turtle carries hundreds of defective mutations in its genome. Those are usually no problem for the health and wellbeing of the individual as the healthy normal variant on the other chromosome compensates for any problems. But when relatives are bred the chances are high they carry the same defective mutations, inherited from their common ancestors - the founders of the breed. When two defective alleles meet in one individual no healthy allele is there to help and the disease is expressed. 150 years of breeding in closed stud books with methods of line breeding led to a about 800 hereditary diseases known in the various dog breeds.
Increasing homozygousity led yet to another serious health problem - the compromising of the immune system. It is vital for the function of the immune system that it is variable and changes from generation to generation to be able to stay ahead of fast evolving bacteria and viruses. A good immune system recognizes threats correctly and distinguishes well from harmless things - like pollen - or beneficial things or the own body cells. Allergies and autoimmune disorders are a sign of a disfunctional immune system. Another job of the immune system is to detect cancer cells and destroy them before they grow into a tumour. Dogs are the mammal species with the highest incidence of cancer known to science. Something is really going very seriously very wrong .
2. Exaggeration of typical traits, overtype, qualzucht
At the same time purebred dog breeding began dog shows became a fancy and starting from dogs who had to be fit for function (companion, herding, hunting, pulling, etc) the main function for those purebred noble dogs now was to enable their owners and breeders to win at shows. As everybody loves the special unique thing dogs got more and more special and different from the normal dog type of a basic canide more and more. In breeds with shortish legs those specimen with the most typical shortest legs won the shows. In breeds with long hair those with the most extravagant ground touching curtain of hair won the shows. In breeds with broad heads and short muzzles those with round heads and extremely short muzzles won the shows. This is called exaggeration and overtype and few breeds are free from it. As one can argue that long hair can be maintained by the owner and have no ill effect to the dog the same is not true for extremely short muzzles, extremely short legs, extreme weak tissue, extreme long ears, extreme skin folds. Dogs with such exaggerated traits live hard lives, some struggle to breath, some have constant inflammation of skin, ears and eyes, some are prone to injury because their biomechanics do not work properly any more because of their freakish proportions.