  We calculate improved lower bounds for the connective constants for
self-avoiding walks on the square, hexagonal, triangular, $(4.8^2)$, and
$(3.12^2)$ lattices. The bound is found by Kesten's method of irreducible
bridges. This involves using transfer-matrix techniques to exactly enumerate
the number of bridges of a given span to very many steps. Upper bounds are
obtained from recent exact enumeration data for the number of self-avoiding
walks and compared to current best available upper bounds from other methods.
