(a) Animals running at large.
(1) No person, who is the owner or keeper of horses, mules, cattle, bison, sheep, goats, swine, llamas, alpacas, or poultry, shall permit them to run at large in the public road, highway, street, lane, or alley, or upon unenclosed land, or cause the animals to be herded, kept, or detained for the purpose of grazing on premises other than those owned or lawfully occupied by the owner or keeper of the animals.
(ORC § 951.02)
(2) Whoever violates this division (a) is guilty of a misdemeanor of the fourth degree.
(ORC § 951.99)
(3) A. The owner or keeper of an animal described in division (a)(1) of this section, who negligently permits it to run at large in violation of that section, is liable for all damages resulting from injury, death, or loss to person or property caused by the animal in any of the places specified in division (a)(1) of this section or upon the premises of another without reference to the fence that may enclose the premises.
B. The running at large of any animal specified in division (a)(1) of this section or upon any of the places specified in that section is prima-facie evidence in a civil action for damages under division A. above of this section that the owner or keeper of the animal negligently permitted the animal to run at large in violation of division (a)(1) of this section.
(ORC § 951.10)
(b) Dogs running at large.
(1) No owner, keeper, or harborer of any female dog shall permit it to go beyond the premises of the owner, keeper, or harborer at any time the dog is in heat unless the dog is properly in leash.
(2) Except when a dog is lawfully engaged in hunting and accompanied by the owner, keeper, harborer, or handler of the dog, no owner, keeper, or harborer of any dog shall fail at any time to do either of the following:
A. Keep the dog physically confined or restrained upon the premises of the owner, keeper, or harborer by a leash, tether, adequate fence, supervision, or secure enclosure to prevent escape;
B. Keep the dog under the reasonable control of some person.
(ORC § 955.22)
(c) Penalty.
(1) Whoever violates divisions (b)(1) or (b)(2) above shall be fined not less than $25 or more than $100 on a first offense; and on each subsequent offense, shall be fined not less than $75 or more than $250 and may be imprisoned for not more than 30 days.
(2) In addition to the penalties prescribed in division (c)(1) above, if the offender is guilty of a violation of division (b)(1) or (b)(2) above, the court may order the offender to personally supervise the dog that the offender owns, keeps, or harbors, to cause that dog to complete dog obedience training, or to do both.
(ORC § 955.99)