For Native Texas Game Animals, there is a limited season for most, as well as bag or possession limits. Those animals and their approximate hunting seasons are as follows:
White-tailed Deer
Archery: End of September through October
General: Early November to early-January
Late Antlerless and Spike: Early January to mid-January

Rio Grande Turkey
Archery: End of September through October
Fall Season: Early November to early-January
Spring Hunt Season: Mid March to early-May

Quail
Statewide: Late October to mid-February

White Wing Dove
Late Summer: Early September through October
Winter: Late December to early January

Pheasant
County by County: October to March

Alligator
Upper Texas Coast: September 10-30, 2011

Javelina
Northern Texas: October to February Southern Texas: Year-round

Mule Deer
Archery: Late September to the end of October
General: Late November to early December

Pronghorn Antelope
By Permit Only: One week in October

There is not a limited hunting season in the state of Texas for Non-native Exotic Animals, nor are there any state bag or possession limits or closed seasons on exotic wild game animals. With a valid State of Texas Hunting License, you may hunt a non-Native game animal year-round. Those animals include, but are not limited to:

Addax Antelope Hunt

Afghan Urial Hunt

African Porcupine Hunt

Texas Alligator Hunt

Aoudad Sheep Hunt

Arabian Oryx Hunt

Armenian Mouflon Hunt

Axis Deer Hunt

Axis Doe Hunts

Barasingha Hunt

Beisa Oryx Hunt

Black Hawaiian Hunt

Black Wildebeest Hunt

Blackbuck Antelope Hunt

Blesbok Hunt

Bobwhite Quail Hunt

Bongo Hunt

Buffalo Hunts

Catalina Goat Hunt

Chinese Water Deer

Chukar Hunt

Corsican Ram Hunt

Dama Gazelle Hunt

Eland Hunts

Eld's Deer Hunt

Elk Hunt

Fallow Deer Hunt

Father & Son Hunt

Feral Hog Hunt

4 Horned Jacob Sheep

Fox Hunts

Gaur Hunts

Gemsbok Hunt

Hill Country Whitetail

Himalayan Tahr

Hog Deer Hunt

Hog Dog Hunts

Hybrid Ibex Hunt

Impala Hunt

Kafue Lechwe Hunt

Kudu Hunt

Markhor Hunt

Merino Ram Hunt

Mexico Dove Hunt

Mouflon Hunts

Mule Deer Hunt

Muntjac Deer Hunt

Nilgai Hunt

Nubian Ibex Hunts

Nyala Hunt

Painted Desert Ram Hunt

Pere David's Deer

Persian Ibex Hunt

Pheasant Hunt

Pronghorn Antelope Hunt

Red Deer Hunt

Red Lechwe Hunt

Red Sheep Hunt

Roan Antelope Hunt

Rusa Deer

Sable Hunts

Sambar Deer

Scimitar Oryx Hunts

Sika Deer Hunt

Silk Deer Hunt

Sitatunga Hunt

Springbok Hunt

Texas Dall Sheep

Texas Dove Hunts

Texas Ram Slam Hunt

Thomson's Gazelle Hunt

Transcaspian Urial Hunt

Turkey Hunt Package

Waterbuck Hunt

Water Buffalo Hunt

Watusi Hunts

White Bearded Wildebeest

Whitetail Deer Hunt

White Elk

White Winged Dove Hunt

Wild Boar Hunt

Wildebeest Hunt

Yak Hunt

YO Ibex Hunt

Zebra Hunts

The Entire Texas Hunting Season Dates by Animal can be viewed online at Texas Parks and Wildlife online here.