In this tutorial, we briefly go through the main functionalities of the library *synaps*. We first start with the main concepts, which underly the design of the library. Next, we show how to construct a first executable, using template structures from the standard library. The next step is devoted to linear algebra functions and the connection to the *lapack* library. Then we describe how to manipulate polynomials in one ore several variables, with different internal representation, including the *Bernstein* basis representation. In the next step, we describe how to use univariate and multivariate solvers. Finally, we illustrate the construction and manipulation of parameterised and implicit curves and surfaces. For each of these steps, we give example codes, that the reader compile and run and also propose exercises.