The purpose of this text is to disclose the premises of the study on the most classic Functional Equations and Inequations, considering their relevance for the development of mathematics, also having the vision of spreading a proposal for research material that contributes to the improvement of teaching in this field of mathematics, which is little explored in Brazilian literature. We present throughout the writing, a brief discussion about the history of some scholars who made use of Functional Equations, showing aspects of solutions for certain standard Functional Equations, namely, Additive Cauchy Equation, Jensen Equation and Linear Functional Equation. Furthermore, we present a detailed study on the classes of solutions that characterize the Exponential, Logarithmic, Functional Equations, Cauchy Multiplicatives and the D’Alembert Equation. We emphasize that we were able to generalize, throughout the work, some Functional Equations, such as the Cauchy’s Additive Functional Equations, with the goal of looking for more complex solutions that satisfy the so-called Pexider and Vince Equations. We also explain a study of Functional Equations involving two variables, such as the Euler Equation and the Additive Cauchy Equation in two variables. We will also discuss certain special cases of a family of Functional Equations of a variable, called the Conjugation Equation, among these, the Schr¨oder Equation , the Abel Equation, and the B¨ottcher Equation. We will also show results on Functional Equations with multiple radicals and Polynomial equations, both proposed by the famous Indian mathematician Srinivasa Ramanujan. Finally, we will illustrate some applications of Functional Equations in Basic Education problems, more strictly, in questions from the Mathematics Olympiads, contained in the most varied events of this category, both in scope national and international.Este texto, tem por amago, divulgar premissas do estudo sobre as mais classicas Equacoes e Inequacoes Funcionais, considerando a relevancia destas para o desenvolvimento da matematica, tendo tambem a visao de difundir uma proposta de material de pesquisa que contribua para a melhoria do ensino deste ramo da matematica, pouco explorado na literatura brasileira. Apresentamos ao longo da redacao, uma breve discussao sobre a historia de alguns estudiosos que fizeram uso das Equacoes Funcionais, exibindo aspectos de solucoes para certas Equacoes Funcionais padroes, a saber, Equacao Aditiva de Cauchy, Equacao de Jensen e Equacao Funcional Linear. Alem disso, expomos um estudo detalhado sobre as classes de solucoes que caracterizam as Equacoes Funcionais Exponenciais, Logarıtmicas, Multiplicativas de Cauchy e a Equacao de D'Alembert. Ressaltemos que pudemos ao longo do trabalho generalizar algumas Equacoes Funcionais, como as Equacoes Funcionais Aditivas de Cauchy, com a meta de buscar solucoes mais complexas que satisfacam as denominadas Equacoes de Pexider e Vince. Explanamos ainda um estudo de Equacoes Funcionais envolvendo duas variaveis, como a Equacao de Euler e a Equacao Aditiva de Cauchy em duas variaveis. Dissertaremos tambem, certas casos especiais de uma famılia de Equacoes Funcionais de uma variavel, denominada Equacao de Conjugacao, entre estas, consta, a Equacao de Schroder, a Equacao de Abel, e a Equacao de Bottcher. Mostraremos ainda resultados sobre Equacoes Funcionais com radicais multiplos e equacoes Polinomiais, ambas propostas pelo celebre matematico indiano Srinivasa Ramanujan. Finalmente, ilustraremos algumas aplicacoes das Equacoes Funcionais em problemas do Ensino Basico, mais estritamente, em questoes provenientes das Olimpiadas de Matematica, contidas nos mais variados eventos desta categoria, tanto em ambito nacional quanto internacional.São Cristóvão