NOVELTY - Non-crosslinked, covalently and/or ionically crosslinked polymers of specific formula are new.    USE - The polymer is useful for electrochemical cells, especially secondary batteries, electrolysis cells, and in polymer electrolyte membrane fuel cells, especially hydrogen and direct methanol fuel cells (claimed), and in membrane separation processes, e.g. gas separation, pervaporation, perstraction (sic), reverse osmosis, nanofiltration, electrodialaysis, and diffusion dialysis (claimed).    ADVANTAGE - Membranes doped with the polymers have a very small specific current flow resistance at 20degreesC, low permeability for water oxygen, and methanol, and extremely thin membranes made from the polymers of total thickness 10-100 micron have outstandingly good material properties, e.g. high mechanical stability at 60-82degreesC. The polymer shows swelling of less than 100% in deionized water at 90degreesC.    DETAILED DESCRIPTION - Non-crosslinked, covalently and/or ionically crosslinked polymers having repeating units of formula (1):    -K-R- (1)    K = a bond, i.e. containing oxygen (2): -C(=O)- ; or sulfur (3): -S(=O)2- ;    R = a divalent aromatic or heteroaromatic compound, where (a) R has at least in part substituents of formula (2I) to (2R):    -R1-S(=O)2-M (2I)    -S(=O)2-R1-S(=O)2-OM (2J)    -S(=O)2-R1-P(=O)(OM)-OM (2K)    -S(=O)2-R1-C(=O)-OM (2L)    -S(=O)2-R1-B(OM)2 (2M)    -S(=O)2-R1-S(=O)2-X (2N)    -S(=O)2-R1-P(=O)(X)2 (2O)    -S(=O)-R1-C(=O)-X (2P)    -S(=O)2-R1-B(X)2 (2Q)    -S(=O)2-R1-S(=O)2-M (2R)    R1 = 1-40C optionally branched allyl, cycloallyl, alkylated aryl, or heteroallyl,    M = H, a metal cation, preferably Li+,Na+, K, Rb+, Cs+, TiO2+, or ZrO2+, or optionally alkylated ammonium ion,    and X = halogen or optionally substituted alkylated amino group, (b) R can have partially substituents of formula (5A) and (5B):    -R2-N(R3)(R4) (5A)    -R2-N(R3)(R4)(R5) (5B)    R2, R3, R4, R5 = H, 1-40C hydrocarbon, preferably optionally branched alkyl, cycloalkyl, or optionally alkylated aryl, at least two of R2, R3, and R4 can be part of an aromatic ring, and/or R = at least partially a group of formula (5C) and/or (5D).    -R2-N(R4)-R3- (5C)    -R2-N(R4)(R5)-R3- (5D)    An INDEPENDENT CLAIM is included for preparation of the polymer in which one or more educt polymers having the functional groups (a), (b), and (d), where (a) and (b) = as above, and (d) = a sulfinate group of formula (6):    -R1-S(=O)-OM (6)    is reacted with a compound of formula (7):    YLm (7)    L = the end product; and    m = greater than 2.