The short answer to this question is that coats of arms were granted to individuals, and not families or family surnames, with very few exceptions. An authentic coat of arms was passed from a father to his eldest son, maintaining its individual ownership. A person entitled to use a coat of arms is known as an "armiger." Historically, the coat of arms was used as a means of identification, to tell friend from foe on the medieval battlefield. At first a coat of arms was designed and adopted by the person using it. Later the right was "awarded" by the Crown for acts of achievement. Over time heraldic design became "fashionable" and was adopted by individuals, independent of battle.
Today, the right to bear a coat of arms be must be granted, certified, and registered by armorial authority. To do so, applicants must verify their legitimate and rightful direct line of male descent from the original armorial bearer. This requires research and documentation, something akin to joining a lineage society. To inquire about this process, you may contact the College of Arms in the UK or the a heraldic authority for your country of interest.
Be advised, there are many companies and websites that will sell you a coat of arms based on your family surname, but such offers are misleading at best. Among some of the top genealogy scams is the selling of fake coats of arms. That said, original coats of arms have been designed and adopted by various institutions such as churches, colleges and universities, states, cities and towns, and even by individuals. There are societies that can aid in the design and registration of personal coats of arms, but again, buyer beware. You may also contact the heraldic authority in your country in which you wish to register.
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