The main object of this paper is to present several properties and characteristics of a number of new subclasses of meromorphically multivalent functions with positive or negative coefficients. The various results obtained here include necessary and/or sufficient conditions for functions belonging to these subclasses. The radii of meromorphically starlikeness and meromorphically convexity of functions in some of these subclasses are also determined. This paper is related closely to the earlier works by O. Altintaş, H. Irmak, and H. M. Srivastava [Panam. Math. J. 5, No. 1, 75-81 (1995; Zbl 0836.30008)], N. E. Cho and J. A. Kim [Int. J. Math. Math. Sci. 18, No. 3, 463-467 (1995; Zbl 0853.30007)], S. M. Sarangi and S. B. Uralegaddi [Tamkang J. Math. 27, 81-88 (1996; Zbl 0848.30009)], M. P. Chen, H. Irmak, H. M. Srivastava and C. S. Yu [Panam. Math. J. 6, No. 2, 65-77 (1996; Zbl 0864.30015)], and S. B. Joshi and H. M. Srivastava [Panam. Math. J. 7, No. 1, 101-109 (1997; Zbl 0867.30014)].