Algorithms are developed for reliable and accurate evaluations of the complex elementary functions required in FORTRAN 77 and FORTRAN 90, namely, cabs, csqrt, cexp, clog. csin. and ecus. The algorithms are presented in a pseudocode that has a convenient exception-handling facility. A tight error bound is derived for each algorithm. Corresponding FORTRAN programs for an IEEE environment have also been developed to illustrate the practicality of the algorithms, and these programs have been tested very carefully to help confirm the correctness of the algorithms and their error bounds. The results of these tests are included in the paper, but the FORTRAN programs are not; the programs are available from Fairgrieve.