Several radii problems are considered for functions f (z) = z + a 2 z 2 + • • • with fixed second coeffcient a 2. For 0 ≤ β < 1, sharp radius of starlikeness of order β for several subclasses of functions are obtained. These include the class of parabolic starlike functions, the class of Janowski starlike functions, and the class of strongly starlike functions. Sharp radius of convexity of order β for uniformly convex functions, and sharp radius of strong-starlikeness of order γ for starlike functions associated with the lemniscate of Bernoulli are also obtained as special cases. 1. Motivation and a survey Let A denote the class of analytic functions f defined in the open unit disc D := {z ∈ C : |z| < 1} and normalized by the conditions f (0) = 0 and f ′ (0) = 1. Let S be its subclass consisting of univalent analytic functions. Thus functions in S has the form f (z) = z + a 2 z 2 + • • •. Gronwall [13] obtained lower and upper bounds for the quantities | f (z)| and | f ′ (z)| for univalent functions with preassigned second coefficient a 2. Corresponding results for convex functions were also obtained. Unaware of these results, Finkelstein [8] investigated the problem again and obtained similar results, except for an inaccurate lower bound for | f (z)|. Corresponding estimates for starlike functions of positive order with fixed second coefficient were obtained by Tepper [41], while for convex functions of positive order, such estimates were derived by Padmanabhan [24]. The problem for general classes of functions defined by subordination was investigated by Padmanabhan [27] in 2001. For close-to-star and close-to-convex functions, such estimates were investigated by Al-Amiri [4] and Silverman [33], respectively. In addition to distortion and growth estimates, Tepper [41] obtained the radius of convexity for starlike functions with fixed second coefficient. This radius result was also obtained independently by Goel [11], whom additionally obtained the radius of starlikeness for functions f with fixed second coefficient satisfying Re(f (z)/z) > 0 for z ∈ D. Following these works, several authors have investigated radii problems for functions with fixed second coefficient; we provide here a brief history of these works.