The main point of this work was to make a generalization of partial recursive functions to partial recursive functionals. First we introduced the notions of computability and a computable function. Then we defined RAM-machines and RAM-computable functions so that we could define a new class of computable functions. This is the class of partial recursive functions. We talked about the main characteristics of partial recursive functions and gave a quick review of the main theorem for partial recursive functions, which is Kleene’s normal form theorem. While we gave the quick proof we also introduced the notion of ”effective coding” of finite sequences of natural numbers by definig the coding function and the code of a finte sequence of natural numbers. We then gave examples of the coding procedure. After we defined everything needed, we introduced the notion of a functional and defined it as a function with parameters that can take not only natural numbers as arguments, but also functions on natural numbers. Then we are stepwise doing a generalization of notions that we had for functions by intuitively giving the definition of a computable functional. We generalize the definition of RAM-machines by defining GRAM-machines which have a new type of instruction. The job of the new instruction is to take care of the function arguments of a functional. Just like the definition of RAM-computable functions we define GRAM-computable functions and say that every RAM-computable function is also GRAM-computable. We then proved the normal form theorem for partial recursive functionals using the analogon of this theorem in terms of partial recursive functions. The next goal, and the last part of this work was to give the proof the important first recursion theorem for partial recursive functionals. We have done that by defining the fixed-point and defining the property of compactness and monothonity for recursive functionals.