For nearly 200 years, the only known complete minimal surfaces of finite topology that were also embedded surfaces were the plane, catenoid, and helicoid.
Costa (1984) constructed a complete minimal surface of genus
one with three embedded ends, and Hoffman and Meeks (1985) proved that the resulting
Costa minimal surface is an embedded
surface. Hoffman and Meeks (1990) subsequently constructed an infinite family
of three-ended embedded surfaces, one in every
positive genus. Costa (1993) also constructed a one-parameter
family of complete minimal surfaces of genus one whose
four ends are planar and embedded.
Costa, C. J. "Example of a Complete Minimal Immersion in
of Genus One and Three Embedded Ends." Bol. Soc. Bras. Mat.15,
47-54, 1984. https://doi.org/10.1007/BF02584707.Costa,
C. J. "Complete Minimal Surfaces in of Genus One and Four Planar Embedded Ends." Proc.
Amer. Math. Soc.119, 1279-1287, 1993. https://doi.org/10.1090/S0002-9939-1993-1160295-2.Hoffman,
D. "The Computer-Aided Discovery of New Embedded Minimal Surfaces." Math.
Intell.9, 8-21, 1987.Hoffman, D. and Meeks, W. H. III.
"A Complete Embedded Minimal Surface in with Genus One and Three Ends." J. Diff. Geom.21,
109-127, 1985. https://doi.org/10.4310/jdg/1214439467.Hoffman,
D. and Meeks, W. H. III. The Global Theory of Properly Embedded Minimal Surfaces.
Amherst, MA: University of Massachusetts, 1987.Hoffman, D. and Meeks,
W. H. III. "Embedded Minimal Surfaces of Finite Topology." Ann.
Math.131, 1-34, 1990. https://doi.org/10.2307/1971506.Schwalbe,
D. and Wagon, S. "The Costa Surface, in Show and Mathematica." Mathematica
in Educ. Res.8, 56-63, 1999.
of Genus One and Three Embedded Ends." Bol. Soc. Bras. Mat. 15,
47-54, 1984. 
of Genus One and Four Planar Embedded Ends." Proc.
Amer. Math. Soc. 119, 1279-1287, 1993. 
with Genus One and Three Ends." J. Diff. Geom. 21,
109-127, 1985.