This paper is concerned with connectivity in regular lattices and random mosaics, and makes two major contributions. First, it presents a solution to the hitherto unsolved problems of predicting the expected numbers of connected components in square and hexagonal lattices, using a one dimensional growth approach that can also be used for some other lattices. Second, it investigates the relationship between connectivity in regular lattices and random mosaics. Experimental results are presented for both the cases. (Author)