In this chapter, we consider parametric scalar equilibrium problems and discuss their stability conditions. Sufficient conditions for the semicontinuity and continuity of solution maps under relaxed continuity and generalized convexity assumptions are provided. The chapter not only is an effort to provide a complete and comprehensive presentation of the results obtained so far but also contains some new facts or refinements of known results. Hölder (or Lipschitz) continuity of solution maps of these problems has been investigated from the beginning, along with several types of (semi)continuity. Combining assumptions on Holder continuity with conditions related to (strong) monotonicity and (strong) convexity, we introduce Hölder/Lipschitz properties of approximate and exact solution maps of the reference problems. The results presented in this chapter could be regarded as an endeavor to unify the corresponding contributions in the literature. We discuss these properties in connection with well-posedness, one of the most important topics in both theory and numerical methods in optimization. Note that, under suitable adjustments, this concept can cover many important ones, including Tikhonov and Hadamard well-posedness. In this chapter, for the results found in the literature, we seek a unified exploration, not repeating the presentation of any article.