The article provides an overview of algorithms in real algebraic geometry, focusing on both old and new developments. It begins by discussing the Tarski-Seidenberg theorem and its proof, which is the starting point for effective quantifier elimination in the theory of real numbers. The complexity of Tarski's algorithm is not elementary recursive, but more efficient algorithms using cylindrical algebraic decomposition (CAD) have been developed. CAD provides an alternative algorithm for quantifier elimination and is more efficient compared to Tarski's method, with a complexity of doubly exponential in the number of variables and degrees of polynomials. The article also covers the critical points method, which is used to find points in every semi-algebraically connected component of an algebraic set. This method involves perturbing the algebraic set using infinitesimals, leading to a bounded nonsingular algebraic set defined over a non-archimedean real closed extension. The critical points method allows for singly exponential complexity in solving problems in semi-algebraic geometry, with algorithms having complexity bounded by $s^{k+1}d^{O(\ell+k)}$, where $s$ is the number of polynomials, $d$ is the maximum degree, and $k$ is the number of variables. Additionally, the article discusses the Block Elimination Algorithm, which eliminates one block of variables at a time, and the Sign Determination Algorithm, which computes the signs of a family of polynomials at the roots of a fixed polynomial. These algorithms are crucial for designing efficient algorithms for real quantifier elimination. The article concludes with a discussion on the computational complexity of these algorithms and their applications in various fields, such as robotics, molecular chemistry, and theoretical computer science. It also mentions open problems and future research directions in the field.The article provides an overview of algorithms in real algebraic geometry, focusing on both old and new developments. It begins by discussing the Tarski-Seidenberg theorem and its proof, which is the starting point for effective quantifier elimination in the theory of real numbers. The complexity of Tarski's algorithm is not elementary recursive, but more efficient algorithms using cylindrical algebraic decomposition (CAD) have been developed. CAD provides an alternative algorithm for quantifier elimination and is more efficient compared to Tarski's method, with a complexity of doubly exponential in the number of variables and degrees of polynomials. The article also covers the critical points method, which is used to find points in every semi-algebraically connected component of an algebraic set. This method involves perturbing the algebraic set using infinitesimals, leading to a bounded nonsingular algebraic set defined over a non-archimedean real closed extension. The critical points method allows for singly exponential complexity in solving problems in semi-algebraic geometry, with algorithms having complexity bounded by $s^{k+1}d^{O(\ell+k)}$, where $s$ is the number of polynomials, $d$ is the maximum degree, and $k$ is the number of variables. Additionally, the article discusses the Block Elimination Algorithm, which eliminates one block of variables at a time, and the Sign Determination Algorithm, which computes the signs of a family of polynomials at the roots of a fixed polynomial. These algorithms are crucial for designing efficient algorithms for real quantifier elimination. The article concludes with a discussion on the computational complexity of these algorithms and their applications in various fields, such as robotics, molecular chemistry, and theoretical computer science. It also mentions open problems and future research directions in the field.