We present the additive arithmetical semigroups and summarize the improvements on prime number theorems and mean-value theorems on additive arithmetical semigroups. We start with definitions and examples, then compare the approaches, which have been used to prove prime number theorems. Thereafter, we give a short outline of the convolution theory and generating functions. Then we proceed with complex-valued multiplicative functions on additive arithmetical semigroups. First we summarize some results for multiplicative functions of modulus less or equal to 1, and more generally for uniformly summable multiplicative functions. Afterwards, we prove new mean-value theorems for uniformly summable multiplicative functions on additive arithmetical semigroups. These theorems are more general than the previous results because our conditions on the additive arithmetical semigroups are weaker and we can prove our mean-value theorems for a larger class of functions. In the proof we use some tauberian theorems by Indlekofer, and some ideas of the proof of mean-value theorems for multiplicative functions in the classical number theory. Finally, we give an application of our results by proving a characterization of finitely distributed functions on additive arithmetical functions and the Three-series theorem on additive arithmetical semigroups.